OK, so good afternoon, everyone. So after a couple of lectures of quite intense background and motivation, I think now we are ready to get into the technical part of the course and start proving some results. So throughout, we will always have some space or some set, metric, let's say compact metric space. We will always have the Borel sigma algebra. And we let m equals the set of all mu Borel probability measures. Then we also have some map, which in several situations will be continuous or differentiable. If this is actually also linear, has some additional, is a manifold, a differentiable structure, certainly at least we require it to be measurable. That just means that the pre-image of every element in the Borel sigma algebra is an element of the Borel sigma algebra. So that's just a kind of minimum level of regularity. So this gives us a dynamical system in the usual way. We iterate it. So the purpose of this course is to understand how to use the notation and the formalism of probability measures to understand the dynamics of this map. So first of all, we need to make two definitions. So the first one is we say that mu in m is f invariant if mu of the inverse pre-image of a equals mu of a in the sigma algebra. So in a moment, I will give several examples and we will discuss this. Notice that the fact that as in the previous course, the fact that I'm putting f to the minus 1 here does not imply necessarily that f is invertible. What I mean is the pre-image of the points in a. So f minus 1 of a, by definition, equals the set of x such that f of x belongs to a. This is the pre-image of the set a. So easy exercise you can check. It might be strange to see it written in this way. If f is invertible, then this is the same thing. You can just apply f to both sides. And this is the same thing saying that mu of f of a is equal to mu of a. But as we shall see in some examples, when f is not invertible, this is the most natural definition. And this definition says somehow that mu and f are related in some way. So that mu is a particular probability measure that is related to the dynamics of f. We will see in some examples what we mean by this. The second definition is we say mu in m is algorithmic if for all a measurable sets a such that f minus 1 of a equals a, then either mu of a equals 0 or mu of a equals 1. Yes, the second definition. It depends on f because this is a condition on the set a that depends on f. So this condition is true only for sets that satisfy this condition here. So it's saying that if you have a set that satisfies this condition here, it cannot have measure a half or anything between 0 and 1. In general, every set, this is a probability measure. So any measurable set has a well-defined measure that is between 0 and 1. It can be 0, it can be 1, or it can be anything in between. What we're saying here is that a set that is fully invariant, sometimes we call it fully invariant, has to have either 0 measure or full measure. So it depends on f in the choice of the class of sets to which this dichotomy applies. So these two definitions are fundamental to the whole course, okay? And we'll start by just giving some simple examples and then for the rest of the course, we will develop some more sophisticated examples. And we will show how these properties remarkably can help us to study the systems in quite a certain level of detail. So some examples of invariant and ergodic measures, so example one, suppose that f of p equals p. So p is a fixed point. And let measure mu equals the Dirac delta measure on p. So here we have fixed point p that is mapped to itself. Here we have our space, m, our metric space. So I claim that delta p is invariant, f invariant. If once f is fixed sometimes we just say invariant, right? Here this should be f ergodic, ergodic with respect to f as well. It's invariant but if f is fixed sometimes we don't state it again. It's invariant and ergodic. So both of them are fairly easy but let me prove the invariant just so you start getting a feeling of what it's about, right? So to see that it's invariant, to see delta of p is f invariant. Choose a set, let A be a subset of M, okay? So we take a set A. So we need to show this. So what are the possibilities for the measure of the set A? There's only two possibilities. You can have measures 0 or 1, okay? So we just have to check those two possibilities. So suppose first that p belongs to A. Then the measure of A is equal to 1. What about the measure of f minus 1 of A? Exactly, p is a fixed point. So the pre-image of the set A must contain p also. Because if the pre-image of the set A did not contain p, then A could not contain p either, okay? So the measure of the pre-image is also equal to 1 because f minus 1 of A also contains p, excuse me? If there exists only one fixed point, so you see, so I'm not assuming anything about the dynamical system here. I'm just choosing. I'm assuming that there exists one fixed point, but maybe all the points are fixed. Yes, for this example, I'm assuming that there exists one fixed point. But this could even be the identity map, okay? But after I choose one fixed point, I also choose a measure. I choose a specific measure. And I choose that measure to be the Dirac delta on this fixed point. I choose it like that. I'm not proving that there exists, there's nothing to prove. I choose that measure. So once I choose that measure, it's like the rest of the dynamical system becomes invisible to me. Because that's the measure I've chosen. This measure is what I'm looking at in the dynamical system. And this measure gives positive measure only to that fixed point. I don't know what happens outside this, okay? I choose that measure. If there are other points, I will also be able to choose other measures. We will see lots of examples of that. But for the moment, what I'm doing, I'm choosing this measure on this fixed point. And I want to show that this measure is invariant and ergodic, right? So these two definitions, as was correctly pointed out before, depend on two things. Both of these definitions depend on the measure and depend on the map. So you say a measure is invariant with respect to this map if this holds. And a measure is ergodic with respect to this map if this holds, okay? So I choose a measure. In this case, it's the Dirac delta on the point P. I want to show that it's invariant. So I need to show this for every, for every measurable set, right? Often this is not trivial how to do that. But in this case, it's not a problem because we are interested in the measure of these sets and there's only two possibilities. Either A contains the fixed point in which case it's got measure one, or A does not contain the fixed point in which case it's got measure zero, right? In the first case, if A contains the fixed point, we've shown that this holds. Because any set A that contains the fixed point, its pre-image also contains the fixed point. So it has measure one, which is the same as the measure of A. And therefore this is satisfied, right? If P does not contain the fixed point, then mu of A equals zero. But the pre-image also cannot contain the fixed point. Because if the pre-image did contain the fixed point, then its image A would also contain the fixed point, which it doesn't. So mu A equals zero and mu of F minus one A equals zero. So we have checked that this is, okay, so mu is invariant. What about ergodicity? Ergodicity is trivial, because this is in fact two for every set. It's measure either zero or one. So in this particular case, it's trivially satisfied, okay? So ergodicity is trivial. So let's look now at just a very simple generalization of this. Let's suppose that we have periodic orbit, right? So suppose we have Pn equals periodic orbit of period n, okay? So P1, P2, Pn goes back to P1. Doesn't have to have such a nice structure. It can be period one million, like the ones that we have seen in the shift map. For example, 2x mod one, we have periodic orbits of period a million, a billion, whatever, okay? Period n. So what measures shall we put on this periodic orbit? Can I put the measure on P1? Is this measure, so this is the Dirac delta on the point P1. Is this invariant? Good, okay? You can take A equals P1, okay? If you take A containing P1 or A even equal to P1, okay? Then this has measure one and f minus one of A is Pn, that's right. Is contains Pn, yes. And does not contain P1. Well, actually in this case it contains P1, but no, it doesn't contain P1 because the image of P1 is P2, that's right. Contains Pn. Or let's just put it this way, does not contain P1. So the measure of A is one and the measure of f minus one of A is zero, not invariant. So how do we put an invariant measure on here? Can we have an invariant measure on this periodic orbit? We need to have some measure on P1 and some measure on Pn, right? And some measure on each of the points along the orbit. Take the average, very good. So we can take the measure mu equals to one over n delta P1 plus delta Pn. So we leave it as an exercise, same exercise as this to show that this is invariant. What about ergodicity? Is it ergodic? Why is it not ergodic? So let's remember that ergodicity, let me write down the definition.  It is this implication, is ergodicity, right? The fact that this implies this, this is the definition of ergodicity. So it looks like this measure is now distributed in various different pieces. So maybe it's not ergodic. But let's check. We need to check. So to show it's not ergodic, all we need to do is find a counter example. So find a set that satisfies this that has measure between zero and one. So proving non-ergodicity is often easier than proving ergodicity. So let's prove, either non-ergodicity or ergodicity. What set are we going to use? Shall we take a set like this that contains only P1, set A. This set has measure between zero and one. What's the measure of each point here? One over N. No, exactly. PN or shall we have both? Like this, is this good? Ah, we need to contain all the points. So now what's the measure of the set? There's a problem. What's the problem? So the only way, if you have any set, so if you have a set that does not contain any of these points, then its pre-image will not contain any of the points. So it will have measure zero, okay? Whether it satisfies this or not. But if it contains any of these points, then either it contains all of them, in which case it may satisfy this or not satisfy this anyway, it's got measure one. But if it does not contain all of them, then it cannot satisfy this. So it is ergodic, because every time it satisfies this, it must satisfy this. So this shows also how ergodicity depends on the map and not just on the measure. So the measure is indeed, in some sense, intuitively divided into some separate bits. But the map means that these bits are connected in a way that you cannot separate them. They are intrinsically connected to each other. So mu is f invariant and ergodic. Okay, so let's look at some more examples. So example three, fm2m is the identity map. Always check the identity map as an example or counter example of everything. So what are the invariant measures for the identity map? So the identity map, this is x, okay? x goes to x. Every point is a fixed point. So what are the invariant measures? Remember, so this is ergodicity. What's the definition of invariance?  So if you have the identity map, what's f minus 1 of a? It's a. So if f is the identity, f minus 1 is also the identity. So f minus 1 of a is equal to a. So the actual sets are the same for any set, which means that any measure you take, this will always be satisfied. So for the identity map, every mu in m is f invariant, every measure, okay? That's a lot of measures. The measures you have for every point is fixed. So you have a direct delta on every measure. You have all kinds of combinations. You have many measures in general for this. What about ergodicity? That's correct. And why is that? So you said that it's ergodic. So mu is ergodic, if and only if mu is the direct delta of some p. Because the measure, one way we just did it before, right? If it's a direct, every point is fixed for the identity. So if you take the direct delta on the fixed point, then it's ergodic. What about non-ergodicity? Why, if it's not like this, why is it not ergodic? If somehow the measure lives on more than one point, why is it not ergodic? Well, for example, let's just try first a simple conversely. So if let mu equals 1 half of delta p plus delta q, where these are two points, why is this measure not ergodic? That's right. Or you can just take any set containing p. It satisfies this condition because it's the identity map. And its measure is equal to 1 half. So it's not ergodic. So in general, because the identity map is so, every set satisfies this for the identity map, then clearly any measure that gives positive measure to more than one point, you can split that set into two parts where both parts have positive measure. Because you can just take any set that contains more than one point, you can divide into two sets. One that contains one point and one that contains all the other points, for example, or somehow you can divide into two sets. So in this case, if mu gives positive measure to more if, let me say it a little bit differently. So if mu is not of the form delta p for some p in M, then there exists sets A and B in M with measure of A positive, measure of B positive, and A intersection B is empty. The only way that this cannot happen is if mu lives completely only on just one point. And in this case, you just need two sets with positive measure because every set satisfies this is automatic. This argument will not work in general. This statement is true in general. But in this case, since f is the identity, f minus A of A equals A. And so, and this gives a contradiction to a good density. Another couple of examples. Example four, a favorite example, f, 0, 1, 2, 0, 1, x goes to 2x, 2x, so 0, 1. So what invariant measures do we have for this map? Excuse me? Excuse me? So the only measures that are organic for the identity map are the Dirac deltas on a fixed point. So every measure is invariant. So in general, if you take the identity map, for example, on the interval, the interval has lots of measures. Remember, we did the examples last time. It has Lebesgue measure. It has all the measures that you get by putting some density, integrating some continuous function with respect to the Lebesgue measure. It's got the delta Dirac on the fixed points. It's got, well, on the fixed points, on the periodic points. It's got lots of measures. But in this particular case, every measure is invariant because every set, the pre-measure of every set is itself. So every measure is invariant. Now we ask, which of these measures are organic? If you take the Dirac delta on one fixed point like this, then it's organic because we showed it before. Remember, if you have a fixed point, the Dirac delta on that fixed point is organic. The question is, if you take another measure, for example, if you take a measure like this that is leaving half of it on one fixed point, half of it on the other fixed point, then it's not organic. You can see that. This measure is not organic because you can just take a set here that contains only p. This has measured 1 half.  But you can generalize this fact to say, OK, any measure that is not of this form must necessarily, because it's not fully concentrated on a single point, you must be able to find two sets to disjoint sets, both a positive measure. If you could not find two disjoints on a positive measure, then it would exist only on one fixed point. If it somehow exists on more than one fixed point, then you can do that. And because each of these sets always satisfies this, then that means you found two sets that satisfy this and the measures between 0 and 1. So it's not organic. So no measure is organic except the Dirac delta on the fixed points for the identity map. Yes? Any measure what? I don't understand your question. No, no. So, or you take measure only in 0, 1. Any measure where the only possible measure of any subset is 0 or 1 I think so, yes. Yes? Well, but how can a measure take just values in 0, 1? Because if, yes, it has to be the Dirac delta measure, yes. Otherwise, yes, otherwise it needs to take, there must be some set that has measure between 0 and 1. That's right, yes, I guess that's what, yes. OK, what about this map here? It has some fixed points, right? So it has the fixed point at 0. So we can have the Dirac delta on 0. It has some periodic points. Remember that it has periodic points? It has many periodic points. So each of these periodic points admits the average of the Dirac deltas along the periodic orbit that we, the example we gave before. Are there any other measures that might be invariant? There's really only one that we have studied specifically, that we know specifically, which is a candidate, which is, for example, the Bayek measure. These are the measures that we know so far, OK? So is the Bayek measure invariant? Is the Bayek measure invariant? Well, let's look. Let's take a set A. Let's look at an open interval. If the set A has a certain measure, the Bayek measure, what is the pre-image of the set A? Exactly. So the pre-image of the set A will have two components. What is the Bayek measure of this pre-image? Each one is measure half. Half of the measure of A, of the length of A. So the union of the two has the same length as the interval A, the union of the two, OK? So at least for these intervals, the Bayek measure of F minus 1 of A equals the Bayek measure. So strictly speaking, to check invariance, we need to check it for every Borel set, OK? But if, again, using a kind of argument that I won't go into but using the fact, the Kolmogorov extension theorem, this is one of those examples where it is enough to check this property on the algebra of unions of intervals, OK? And then if you check a property of this measure on this, then it automatically, by the fact that this measure extends uniquely to the Bayek measure, then this property also extends to the sigma algebra generated by this algebra, OK? A very fancy way of saying that in this case, or certainly at least as far as this course is concerned, it will always be sufficient to check certain properties for open sets or for intervals in this case. And these imply the same properties for all the sets in the Borel sigma algebra, OK? It's a technical detail that I don't want to go into, let's say, OK? But it follows by the uniqueness of the extension of the measures. So in this case, you can easily see that for intervals, it's clear that we have this property and therefore it satisfied for all sets in the Borel sigma algebra, and therefore this implies that the Bayek measure is F invariant. What about ergodicity? So in this case, ergodicity turns out to be true, but it is not something that you can just see. This we will prove later on in the course that the Bayek measure is ergodic, but it's a non-trivial proof. And we need some dynamics, and we need some interesting argument. And yes, then will it work? If we have x equals kx of 10x, 1, 2, 3, 4, 5, 6, x goes to 6x mod 1. Is this what you meant? Yes. So is the Bayek measure invariant for this case? Yes, by the same argument. So this is, in fact, an interesting observation. So just like one map can have many invariant measures, the same measure can be invariant for many different maps. It just happens to be the Bayek on the interval. It is a special measure, but there are many different maps for which the Bayek measures in Bayek. So ergodicity is true, but non-trivial. We will prove it later. And let me now give you one final example in which we prove. So as I said, sometimes non-ergodicity is easier to prove. And I will try this example to show you why ergodicity. The first reason why ergodicity is actually an important property, number 5. So again, let's take f, 0, 1. And let me draw the graph. So is the Bayek measure invariant in this case? What is the slope here of these graphs? What's the slope of this? 4? Look, this takes an interval of size 1 quarter. And it maps it to an interval of size 1 half here. What is the slope of this? Minus 2, because this interval of size 1 half goes to an interval of size 1. And this is 2. Plus 2, minus 2, plus 2. It's always 2. So let's take an interval here, a. What is the pre-image of this interval here? It will have two components, one here and one here. So what is the size of the union of these two components? It's the same as the size of a. What if we take it somewhere different? If we take it here, b. Then it will have two components here with the same topity, also here. You can check if you take it in the middle. Also you would get the same thing. So you always get invariants. So in this case, the Bayek is f invariant. What about ergodicity? Come? Why not? You take a small interval here, 0 epsilon. OK? Yes, but you can always find a set that is measured different from 1. That's not, be careful not to make the mistake, right? In the identity, I emphasize, you can also here. I can do the same thing. Also here I can find an interval that has positive measure. If my invariant measure is Le Bayek, of course I can always find two sets of positive measure, these joint sets of positive measure, OK? But what property do these sets need to have? Remember, what's the definition of ergodicity? Yes, but not just 0 or 1. There's only specific sets with certain properties should be constrained. The definition of ergodicity, if you take a set with a certain property, then it is not allowed to have measure between 0 and 1. It has to be 0 or 1. But what is the property that the set has to have? Right. So is that true for this? What is the pre-image of 0 epsilon? The pre-image of 0 epsilon is here. So it's not equal to the set itself, right? So to check non-ergodicity, if we want non-ergodicity, we need to find some set A such that F minus 1 of A equals A. And we need to find such a set that has this property whose measure is strictly between 0 and 1. Or we can try to find it here.  Where's here you can? What's set? You need a set. Yes, but you've bought these sets that you've, both of these sets satisfy this, but they both have said it measures 0 or 1. So it's not good enough as a count, for example. So far you've just confirmed ergodicity. You have to look around here, because this maps, no, because you see this maps over itself. It stretches. You need to think more global. You know what? I drew the wrong graph. Sorry about that. Sorry about that. No wonder you could not find it. Sorry. This is the graph it needed to be. If you can tell the difference between this and the other one. Yes. Yes, sorry about that. But it was useful, maybe, also to see the difference between the two. So what's the right set in this example? 0, 1, 2, 1 half. 0, 1, 2, 1 half. The interval 0, 1, 2, 1 half. Very good. So if we take the interval a equals 0 to 1 half, then you look at the pre-image of this, 0 to 1 half. The pre-image is exactly 0 to 1 half. No, I don't understand what the problem is. 0 to 1 half. 0 maps to 0. 1 half maps to 1 half. Ah, here. You're right. You're right, you're right. Yes, yes, yes. Yes, yes, you're right. There is a little problem here. So let me define this on the open interval. Let me define this on the open interval. Yes, yes, well done. So this set satisfies f minus 1 of a equals a. And the big measure of a equals 1 half. Yes, yes. No, that's right. Because exactly the only way, this lack, because actually it turns out, we shall see again that if you just now restrict yourself to this part, then this is ergodic restricted to this, and this is ergodic restricted to this. So the lack of ergodicity means that really for that measure, you have two dynamical systems, separate dynamical systems. That's what this lack of ergodicity means. Is that whatever's happening dynamically here is completely independent of whatever's happening dynamically here. The two systems are really two. If you want, you can change this. You can make a perturbation here. And this system will not know about it. So the lack of ergodicity means that the measure has two different components. In this case, the ergodic components, as we shall see of the measure, one ergodic component is Lebesgue measure restricted to 0.5. And the other components is Lebesgue measure restricted to 1.5.1. And each of these components is ergodic for the map. So really what we have is the non-ergodic measure split into two ergodic components. And as we shall see, that is a more general feature. Whereas in this case, because the interval 0.5 is mapped to this interval here, and this one is mapped to here, and the dynamics on each of these is similar to this one here. In fact, in this case, we do have ergodicity, and we cannot decompose the system into two different parts. OK? So after these examples, what are the theorems? So these are just some examples. In principle, we don't know whether any invariant measures exist or whether any ergodic measures exist. So let's state the first theorem. We want some general theorem about the existence of invariant and ergodic measures. Let's define Mf is equal to the set of probability measures, such that mu is f invariant. And E of f equals mu in Mf, such that mu is, so just a brief remark here is that in the definition of ergodicity, I do not assume that f is invariant. So the two definitions are independent of each other. A map measure can be invariant without being ergodic, but it can also be ergodic without being invariant, OK? Although in practice, at least in this course, we will really only be interested in ergodicity of invariant measures, OK? So generally, we're looking for invariant measures, and then once we know that a measure is invariant, we're interested if it's ergodic. So clearly, we have these are all subsets Ef. This is just a definition. So Mf is the set of probability measures on our space, which are f invariant, just the definition of a set of measures. It may be empty, OK? Ef is a set of invariant measures, which are also ergodic, invariant and ergodic, because I'm assuming that I'm in mu f. Yeah, maybe I should say just mu f ergodic for a given dynamical system. So I'm always assuming here that our set is given with this Borel sigma algebra, and the map and the measure is given. So the theorem, let M compact and f, M to M continuous. Then Mf is non-empty, convex, and compact. Mu, moreover, mu is ergodic if only if mu is an extremal point. So I will recall some of the definitions here. So this is several statements, OK? So the first statement is that mu f is non-empty. What does this mean? Not that there exists a delta measure, just that there exists a measure, a probability measure that is invariant. For example, there are systems with no periodic orbits, no fixed and no periodic orbits, right? What examples have we seen of systems with no fixed or periodic orbits? Which have no fixed orbits and no periodic orbits? Sorry? Yeah, the adding machine, very good. Symbolic system, adding machine where every orbit is dense, and also circle rotations with no fixed or periodic points. Circle rotations have Lebesgue measure that is invariant, because they're just translations. So if you take an interval and you translate it, the big measure is invariant. But just to emphasize that this non-empty is non-trivial, because if you have a fixed or periodic point, then it's trivial, because you can always take at least that delta. But in general, this might not be the case, OK? So the fact that if f is continuous, this is non-empty is non-trivial. In fact, the proof of it is very interesting. We will prove this, OK? Convex. What does convex mean? Let me just remind you. I'm sure you all know. But just for let me recall, mf is convex. So what does a convex that mean? If for all mu 0, mu 1 in mf, and for all t in 0, 1, the measure mu equals t mu 0 plus 1 minus t mu 1 is also in mf. So every linear convex combination belongs to the set, OK? That's what it means for the set to be convex. This is easy to see that if these two are invariant measures, then this measure is invariant. It's easy to just check from the definition, actually. So compact, what do we mean by compact? Compact, we need a topology. What's a topology that we mean by compact? Week start topology, OK? So I don't need to recall the definition of compact. I just want to remind you that we mean compact in the week start topology. Because there are other topologies. But this is what we mean by compact. Finally, what is an extremal point? So what this is saying is that within the space of invariant measures, the measures which are gothic are exactly the extremal points of this set. So a point x is extremal. So mu is extremal whenever we write if mu equals t of mu 0 plus 1 minus t of mu 1 implies t equals 0 or 1. So an extremal point is one that you cannot write as a linear combination of two other measures, of two other points. That's the definition of an extremal point because it's like imagine that it is on the edge in some sense, OK? These are just standard definitions. So in the next lecture, we will prove this theorem. It's the fundamental theorem that says that this sets a non-empty under very general conditions. Today, I just want to give a corollary of this. So corollary. Again, m compact continues. Then e of f is non-empty. And there exists unique probability measure mu hat on n such that 1 mu over e of f equals 1 and 2 and integral e f integral. Sorry. Yes. T. Let's see a little bit what we have just said. So yes. So this mf has a topology, the weak star topology. Therefore, it has a sigma, the Borel sigma algebra associated generated by this topology, OK? Therefore, we can define a measure on this set. So I will prove it in the sense that this is a trivial consequence of some very deep results in functional analysis, which I will state in a second. We will not state those results, OK? But I want to explain a little bit what this result means. So first of all, what does the statement mean? So yes. So the first statement is that e f is non-empty. And the second statement is that there is a probability measure on this set of invariant measures. And this probability measure is supported just on the ergodic measures, OK? And what this is saying is that for any invariant measure, in particular, a non-necessarily ergodic measure, if we want to integrate a function with respect to this measure, we can integrate it as a kind of a product integral. So here you're integrating with respect to this measure on the set of ergodic measures, because this measure lives on the set of ergodic measures. You're integrating the function phi with respect to all the possible ergodic measures, OK? This nu belongs to the set of ergodic measures. So for each ergodic measure, you take this value here. This is a function of the ergodic measure. And then you integrate with respect to this, which means you're taking a kind of average of the integral of this overall ergodic measures according to this mu hat. So what this is saying is basically that what we would like to say is that every measure is a linear combination of ergodic measures. Because you see, this says that the ergodic measures are the extremal points. And you would like to be able to say that every measure is just a linear combination so that every non-ergodic measure, like the examples we saw before, are just the average of two ergodic measures or many ergodic measures. Like remember before the example, we had two fixed points and we had half delta p plus delta q. Each of these is ergodic, but we take the average of them and we get a non-ergodic measure. So we'd like to be able to say that that's true, but this is not necessarily true because it could be morally a linear combination of an uncountable set of ergodic measures. For example, if you take the identity map on the circle and you take Lebesgue measure, remember for the identity map, the only ergodic measures are the fixed points, the Dirac deltas. But every measure is invariant. So if you take the identity map, say on the interval on the circle, then this is the set of all ergodic measures, sorry, all measures, all probability measures are invariant. The ergodic measures are just the Dirac delta. So we'd like to say that every measure, like for example Lebesgue measure, is a linear combination of ergodic measures. But it's not a linear combination of a finite number or even a countable number of such ergodic measures. And therefore this is the generalization of saying that the Lebesgue measure is the average of all the ergodic measures in some sense, weighted by this, okay? So of course this measure here depends on the map, F, obviously, right? So for a given map, it says that you can write any invariant measure as some kind of integral of this on the ergodic measures. So let me state a little bit what are the feelings that this follows from. So the corollary, so the truth of corollary is the Krine-Millman theorem. I'm not sure if you did this in the Gaussian functional analysis, but the Krine-Millman's theorem says exactly that if Mf is non-empty compact and convex, this implies that it is the closed convex hull and Mf is the closed convex hull of its extremal points. So in particular, if it's non-empty and it is the closed convex hull of its extremal points, in particular it must have extremal points. So in particular, Ef is non-empty, okay? Because by the previous theorem, this is the, these are the extremal points. And the second statement, so N1 and N2 are exactly showcase theorem for non-empty compacts, convex for non-empty. So these are really the formulations of general functional analytic theorems in the context of the specific set of probability measures that we have here. Okay, so I think we can leave it at this for today and the next lecture we will prove theorem one. Okay, thank you.


Hello everybody, Andrea here with dental L so have you been audited so your CDHO number if it ends in four or nine you have been audited okay. So let me help you guys, it does not have to be stressful, so if you want to join me February, what did I say February 7 which is a Sunday. i'm going to have a full session on the practice profile i'm going to have a full session on your. Learning activities and how to write goal statements Okay, so this will be big, so this is for those of you who are looking to write your portfolio, so when you are audited you have two choices, you can write your portfolio. submit it, or you can take the quality assurance exam Okay, so this will be specifically for those of you. Who want to write the portfolio and submit your portfolio, but this will also be for everybody, the first portion. We will be talking about the practice profile, it does not have to be stressful, so how does this work, so the first portion. i'm going to talk to you guys about the practice profile because this applies for everybody every dental hygienist has to submit their practice profile. But you might be very confused on how to go about that it does take preparation, so I go specifically through. All of the short answer questions I go through all of them actually, but I talked to you guys about the short answer questions first, because those are what most hygienists have trouble with. And there is some preparation that will need to be done if you realize you're not doing everything you should be doing for best practice. So I go through the dental unit water lines, I go through handpiece maintenance. I go through sterilizing and I go through what do you do if your sterilizer is recalled, so those are the main short answer questions that I go through. Plus I go through if you work in multiple practices because your practice profile will need to be written in a certain way, if you work for multiple practices. So that will be our first session practice profile, and then if you want to join our next session that day that will be on specifically writing the portfolio so you're you're learning activities how to write goals what makes a good goal what makes a not so good goal and you'll have to change that. I go through your learning activities and how to write bibliographies for them, depending on what the learning activity is from so if you're taking webinars or. Listening to them if you're reading your textbook if you're taking online courses I go through how to write out your learning activities. bibliography perfectly, so you don't get that phone call from the CDHO, but I also go through goal statements what makes a good one, what makes a bad one. I go through your reports on learning so in that area, you have to write down what you have learned. Since taking whatever course your goal is from plus what changes you have made to your practice, so I go through both of them. I go through what makes good report on learning what makes bad ones and what the CDHO wants to see to pass your audit, so this is a bigger one where we go through everything. You can have your portfolio done and your practice profile done before the summer before. Can I even say COVID lessons things a little bit so we're able to go out get things done while you're in lockdown, so you don't have to worry about it later. Trust me on this this happens every year, you have your best of intentions to be organized, but then you're not organized and then come Christmas time you're like Okay, I have to get this done i've been putting it putting it off putting it off. Get it done now Okay, so this will be for everybody who you know we all have to write our practice profiles every year, so that will be the first live class the second live. class will be about your portfolio, how to go through all of that Okay, plus there will be a course for both so it's not just a session where we talk about it, if you cannot attend live. There will be a a session recording that i'm going to upload inside your courses, and this will include support by me, so I can help you guys answer questions. We all probably have this you know you all probably have the same questions i'm going to go through all of that with you. So get it done in February, so you guys don't have to worry throughout the year Okay, I will leave the links for you guys at the bottom. super easy to sign up you guys will have everything and full support by me plus there is also an option for me to review your practice profile. And review your portfolio when you're done to make sure you're on the right track, so thank you guys so much for listening, I will hopefully see you guys in February see you very, very soon.


What a glorious past, full of contributions to business, to science, to technology, to the arts. And what an amazing future, in so many ways India, despite its centuries of tradition, and contribution, is a young country, full of promise and hope. Indians and Albertans are naturally aligned with the potential to create incredible prosperity for people in both places. I'm delighted that the Indian Consulate in Toronto, in association with Sanskrit Party Canada, is launching a mobile Sanskrit library, as well as the little guru app. These initiatives will go a long way in popularizing Sanskrit language in Canada, and it will also encourage younger generation of Indian community in Canada to stay connected with their rich culture and glorious heritage. During the pandemic, when we saw members of the Indo-Canadian community helping out other Torontonians and helping those abroad through vaccine donations. And on behalf of the City of Niagara Falls, we'd like to offer India and all of those celebrating congratulations on 75 years of independence. Best wishes. Canada and India share a strong, proud and enduring relationship that thrives in communities right across our nation. It is my hope that the bond between Mississauga and Indian communities continues to grow.


so much man if you haven't already shown some love appreciate it zanzibar welcome to the subs dude i'm gonna run around this trap body because i can we're sub neighbors oh i'm gonna tell what is this a claw a plate they were so shiny for a second there i thought it was a gem i got a tail though a tail webbing hide a lot of rares on this one i like this quest it's not a gem because i'm gonna eventually need a gem and i quoted myself saying it's going to drop it's going to drop you remember you remember me saying that chat i remember me saying that i said that out loud i was like it's gonna drop this is something that's that's that's going to happen it's it's going to drop remember i said that i remember saying that bull wait oh i don't have any um traps shock trap not shock traps without flash flash i'm down to 115 flash bugs oh no that's bad oh has it been five yeah all right let's do a sub giveaway it has been five subs so we're doing a sub giveaway thank you for the reminder uh giveaway there we go giving away a sub to this stream and if you already sub you can enter as well and if you do win and you already sub i will get you a sub to whatever channel you want just give me the channel name if you win sub goal admit we got a goal of five new subs today we got it so i'm just setting up this giveaway real quicker first ticket free same confirmation ticket cost is one point let's do it oh we're gonna let this run until the end of the next hunt all right investigations that was this one right yeah yeah yeah wait there we go that was such a good movie for his time dude i am a walking movie reference i don't remember what i said which movie did i quote i don't even remember there's so many of them that's the most important session in chat if you want to join us oh flash i don't know why i'm doing a badger badger dance maybe it's good for my exercises all right let's go gotta pay attention and actually trap the heck why you know put me on camp s'mores hit the button everybody's throwing something over the chat 36 months in the row billion it's been three thousand years three thousand thousand thank you so much for the support dude i saw the quest half way across the map from the camp and i didn't know what was going on they just dropped you off in random spots man it's like what was it it just is just another level of rng you have to deal with i hate it too man i really do i was like why did you end up having because i actually spawned right next to the monsters i had to go all the way back to uh here oh there he is oh no that's i am i wish i had that i should equip it oh i guess i went the wrong way did he he jumped okay he's down oh hello what there we go no that's not it i don't know what the button is right click there we go we do i've got flash bars now y'all in trouble actually here we go right here i am your savior i don't i don't what happened i didn't put flash flowers on this loadout that's what happened i didn't put them on this loadout i was like what's going on okay so now we're gonna r r v we're ready he i don't i don't particularly like you so i'm gonna put on mantle i look like i'm doing that that good a damage but i've got a lot of things i'm doing wrong stop shooting him stop shooting him oh well oh flies thank you yeah the keyboard is just awkward with bill i'm trying to keep it consistent so i can stay in line with all the melee weapons too that way my brain can adapt to it better if i keep changing control schemes depending on weapons it's gonna take me twice as long to get it so i'm doing my best to stick with mostly default stuff and then um yeah can i craft a trap no i'm out of that stuff he's pretty much ready to be captured though and i'm out of traps not anymore oh wait i thought i thought that was the thing i needed the toaster now i see why dampness does not have ranged weapons why you think that because of a poor pc board on monster hunter no i don't think that's the case i think don't this doesn't have ranged weapons because of balancing i don't think it's controlled i think it's balancing there it is i called it i called it i called it called it called it i called it i called it no they are supposed to they said they were going to add ranged weapons um this oh freezer sack i needed that didn't i yeah oh no some kind of cold sack um they were supposed to have the ranged weapons in august but they have updated the roadmap i think yesterday and it looks like they're not going to make august for the dual repeaters i need the low rank sack crap man i got the gym so i guess that's good um so it looks like the repeaters are not going to make it to august so september for the dual repeaters and on damas i guess all right let's let's get a low rank oh you are too good to me we got the gym though that was the hard thing uh leji on a five will count right i think five will count yes six of mine thank you okay it does have it all right i was wrong uh where are the flash straps now we can save this again bull so we'll do the five star one oh it's two no no no no we don't want that one no no no no no no no no no i'd rather go lower than get two monsters is that all i have really i sort it by monster and all i got was high rank ones i don't want that one you put one up thank you get him stand by stand by for titanfall oh i was supposed to draw for the winner hold on let's draw for the winner the winner is uh kairou kurosaki kairou kurosaki congratulations you have just won a sub to my stream you sure did with one ticket pretty sure everybody got one ticket though but congrats man let me do that right now gift subscription uh why isn't this scrolling down tier one because it's dumb i'm just gonna let the chat scroll some more so i can actually do this because twitch is interface is stupid i can barely see the button there we go i'm just trying to click on your name and chat and do it congrats oh yeah and that's for hitting our sub goal thank you so much guys giveaway is complete enjoy those evo's man you got them for at least a month hey magama what's up man yeah we got a lot of hosts today man it's awesome it's like wow i got like three it was like right back to back man these guys gonna kill she's pretty good overwatch solid host i gotta check out her stream i watch overwatch a lot oh you better not leave what's he doing i just walked right in the stank you better not jump back down here once i get back up here hello hello you're just like ice everywhere up here i'm gonna yeah that's gonna hurt i didn't even eat i just realized uh we're support i'm officially support i didn't eat i got caught up in the giveaway i'm sorry uh i'm coming back up boys i'll be there in a second hey one of the ways to stop it down again i'm trying to adjust my reach uh it's in two weeks it's not this Saturday it's a Saturday after that the best way to remember it is that i do i do something that is the first Saturday of every month the first Saturday of every month so it'll be the first Saturday in september crap damage yeah cool man i'm glad you're doing that glad you're doing that i'm kind of getting hang of the controls now um i'm on a quest to learn keyboard and mouse with the uh the bow it's very very awkward i don't know where he is well he was there it's very awkward especially when you have to scope in with the default controls like i i was playing with a heavy bow gun and i remapped the scope to right click but it messed up like a bunch of other stuff especially if you switch to melee i can't mount hey closer there we go first time i used bow i couldn't aim i have to use the lock on like the zoom in there's the default keys are c and v on keyboard mouse so you have to uh you have to do that and then it becomes easy yeah i had trouble with the the bow on console i was missing even with the lock on i was missing you're not even gonna change my time we get up there i need to i need to get that thing so i can swing on the bugs whatever it's called i've never used that and i see people just like turn the spider man yeah there's lock on just well there's actually there's an actual lock on and then there's scoping in like my key to lock on is some weird key because you can like lock onto the target i don't like that because i don't like i like having more control over my camera than that um but the uh to scope in you c and v on keyboard mouse all right we're gonna support i'm coming i made it i need this freaking jelly whatever it's called for this uh ice bow it's f by the fault i think so i remaped it i think i ate the frost sack let's hope i get it from rewards right now i'm using the uh the diablo's bow and i'm hoping to get the legion ice bow and we'll get a firebow too that's gonna be fun i don't like fighting any of the fire monsters none of them they're all bad i don't like it don't like it at all dance with me come on dance with me baby come on where's i don't see the sack what's oh my goodness i got all this deliciousness and not one of them's a sack come on game i've been farming for exactly 20 minutes one would think you would have a sack by uh angina for fire oh actually angina is not that bad of a fight angina is not that bad um mr bones uh tobey for thunder and a legion for ice hey gorge what's up man welcome i had to capture to get sacks and we might have to cap then you have another one uh minus the post if you're having already because i don't have any low level uh legion stuff all my stuff's high level for some reason yeah cap will up the chances a little bit i'll submit it i almost said post i had to think about that it'll cap this time all right i'm gonna sit down congrats again man on that sub winning that sub and the giveaway make sure you use one okay whenever you can and any channel you go to that's a requirement that's a requirement oh let's go uh sorry okay this time i won't forget to eat yeah tell you boy they got some they got some loony tunes running boy especially are you running for the monster they like leaning back running like i didn't mean to do that are you gambling all your points away man what thanks camera now i'm dizzy wait it turned and looked down oh that was correct where is he is he moving around it only does that when he moves around flips the camera around for no reason drives me nuts okay here we go we're on the right track now pun intended some more points so i can get more investigations here there's a little bit of a cool down the east commands how that not get hit by that i want to go too hard on them you gotta cap them have a mask for it no berries for this right i hate falling off here because you have to go like all around the world to get back up at least i got my mantle i just broke it what is he doing you use the wedge beetles to climb faster i know i don't have the uh what you need a skill for that equipped i forgot what this called it's like a skill you can equip you can just go whoop whoop and no do those things like that but i don't have it equipped i need to equip it on this level especially just aim the slinger was well it didn't work oh crap i'm late there we go train bombs i think he's just right clicked with the slinger okay i thought you had to have a special skill equipped my dang i'll have to do that next time give me a sec i think i need to actually never mind hercules hercules hercules oh my goodness


Everybody, Matt here for Layton Sports Cards, ripping 15 boxes for Tom C. We have one box of 2022-23, Panini Crown Royale basketball, one box of 2023, Bowman baseball hobby, two boxes of 2023, Bowman baseball sapphire, one box of 2022, Bowman's best baseball, one box of 2021, Panini Spectra football, two boxes of 2022, Bowman from baseball sapphire, three boxes of the 23 Fanatics underwraps baseball jerseys, two boxes of 2023, Tops Chrome Jumbo, and two boxes of 2023, Tops Finest Baseball. All right, again, these are for Tom C. Good luck, Tom. Let's start off with basketball and football, and then we'll rip all the baseballs. We'll keep the sports together, we'll get used to this new setup here. Let's see, this is Royalty, rookie insert to 25, Musa Diabate, five of 25, and let's start off our day with a one-of-one. That is SGA, one-of-one, parallel. Nice. Wow. There you go, Tom, one box, one one-of-one. Nice. Shea Gilgis Alexander, I believe that is the Platinum parallel, I think, and silhouette, rookie jersey auto to 199, Shadon Sharp, nice box here. On the back, rookie jersey, Malachi Brannum. All right, let's do 21 Spectre football. G-Dycott, Xavin Collins, that's to 50, to 99, rookie triple relic, Dwayne S. Gridge, Sam's boy. And to 60, that is rookie dual jumbo jerseys, Mack Jones. To 99, rookie patch auto, Rondale Moore, next pack, pink Nick Chubb, 25, Spectracular, Russell Wilson, that's sweet, that is to 25, and 25, patch, Nick Chubb. Can everybody hear me, okay? What's up, Carl? This is to 60, rookie patch auto, Ian Book, Michael Thomas, and Jamar Chubb. Mark Chase, rookie green scope, this is 23 of 35, 75, jersey, Derrick Brown, and redemption, this is expired, this is an expired redemption, radiant rookie patch signature, Celestial, Elijah Moore, 75, Chubba Hubbard, rookie, Marquise Brown, Dycott, green, that is to 30, nice Hollywood. Who do we call Jake around here, Hollywood? Hollywood. To 99, Marquise Brown, Jersey, and on the back to 99, rookie patch auto, Terrace Marshall, Jr. All right, let's get to baseball, start off with the Bowman hobby. This is first bowman, refractor auto, to 499, and Manuel Bonilla, L.A. De La Cruz, sights on September, Justin Crawford, Ramar, Arias, and Aqua, I'm sorry, I'm sorry, a lava refractor to 399, Christian Serda, Brooks Lee, Tristan Casas, rookie of the year favorites, who picks for rookie of the year right now? Ayo, I want to give you a sheet, man. I don't think he's going to win that. Ayo, he's going to win it. Here, Caminero, and on Hellgann Allen, a topic. And then, you know, that's Carroll, and PG. We're going to be picking someone. Robert Esselt, I guess, Edward. Yeah, I think he would have got called up sooner. He'd have to go on an absolute tear in Coral. Carroll would have to get ice cold. Coral. Parada, Blaise Jordan, Lunar Glow, Roman Anthony, Muller-Gorman, rookie insert, Allie De La Cruz, spotlight, Robert Martinez. Assuming you saw the fight yesterday. Yeah, Miguel Blyce. What are your thoughts? He haven't. Not really, I think. If Jose Ramirez was not, if he didn't have a problem with Tim Anderson before, there's Zach V. Well, if he didn't have a problem with him before that play, he wouldn't have had a problem with that play.  Drew Jones, first moment, Gonzalez. Tim Anderson should definitely train not fighting though. Khalil Watson, Jackson Holiday. Definitely shouldn't. Baez, Miguel Vargas. Which one was better, Batista and Ordoa or that one? That one. That one? Gunner, Sky Blue, rookie to 4.99, Churio. Yeah, you don't see many, well not, don't see many, almost never see a blow-ticket guy off his feet in a baseball fight. Trey Sweeney. I mean, Brady House, Lunar Glow. Joey Batts, the goof, but he took it. This is true. You do that. All right, let's do Top's Chrome Jumbo. Come on, Tom, Vlad, 88. And that's Purple Speckle, rookie auto to 2.99, Joey Weimer. Shane Beiber. Weems and beads. Still lots of room if you'd like a personal rift. Pretty good queue here, but once we get through this first personal break, it'll be a very long queue. David VR, rookie aqua refractor to 1.99, Ramirez and Ramirez. Josh Young, rookie 88, Reese Hoskins, prism refractor. Hunter Green, future stars. Nolan Jones, rookie prism refractor and McCutcheon. Kodai Senga, rookie, Xander and rookie auto. Jordan Diaz. I wonder where he's at in the NL. Who's that? Uh, thank you. Oh yeah. I wonder where he's at, like in the... There's a thing that shows mega sods for awards and stuff. Logan L. Hoppe, Technicolor, Acunia and Rendo. Give me a long second here. It's me, man. They lean too far forward. Alright, Francisco Alvarez, rookie. Morale, there's Nolan Gorman. Javier Asad, prism and Ken Waldechuk, rookie refractor. Fernando Tates, blue sonar to 125. Michael Taglia, rookie prism and Correa. Miggy, Josh Young, Tristan Casas, rookie 88. And rookie aqua wave auto to 1.99, Gabriel Moreno. Austin Hayes, excuse me. Logan L. Hoppe, Jeremy Pena and Titans. Rookie Adley, Gallant. Christopher Morale, rookie 88. Garrett Mitchell, rookie prism refractor and Rhett Beatty, rookie refractor. Alright, last pack of the first Topps Chrome jumbo box. Jordan Walker, Michael Harris, Technicolor. Jonathan Aranda, rookie prism refractor and Rizzo. Alright, Luis Castillo, 88 and rookie purple refractor out of 250. Esteban Kimrose, great legend. Belanger, real fuzzy. Thank you, it was a raise for like one game. Oh nice, one game and they're like yep, we've seen enough. This is Ryan McMahon, orange wave to 25. Mike Trout, let's say a break. Kodai Shingo, rookie. Miggy, there's Moreno, rookie prism refractor and seeker. Francisco Alvarez, future stars. Tatis, prism and Rury. Alvarez, rookie. Christopher Morale, rookie. Nolan Gorman, rookie. Tovar, 88 and Ezekiel Tovar, rookie Otto. Kyle Tucker, dude Tovar is a wizard in the field on the show. Busy? Yeah. Cardinals are playing the Rockies right now.  Corbin Burns, Yorda and Lidello. She's like Jose Iglesias before now. Obviously like the glove first, bent second. Giaccio and whoops, that's two cards. There you go. This is Top Scrum by Phoenix. Game used Jersey, Zach Grankey. Hmm. Popka legend. Busy from a popkin? Yeah, he went to a poppin' eye. Mookie prism, the iversario, rookie refractor. This jacket has a short style. Giacciano, rookie, negative refractor. Adonis prism and Ethan, small rookie refractor. Logan O'Hoppe, Castellanos. This is Will Benson, rookie purple speckle, Otto to 299. David B. R, rookie refractor. He's going to be pretty decent for the rest of Will Benson. Oh, yeah. He's been a good platoon back then. Suzuki Future Stars, Soto Titans and rookie refractor, Oscar Coloss. What's up, Patrick? How's it going? See what Patrick sent home from the national for me? No. There's Jordan Walker, rookie. Wait, what's the win rate? Yeah. There's Gorman, rookie. Yeah. Tyler O'Neill prism and Nolan Jones, rookie refractor. Sixth blue card. Ronald Acuna, tops in technicolor. Doing great here. Jared Walsh prism and Sandiel Contra. All right. Let's do finest. Yeah, man, having a great weekend. How's yours going? I guess you already said Goma. Did you make it over to the shop for the national trading card day? Or the baseball card day? Derek Jeter, rivalries. Oscar Coloss, rookie purple. Just a minute. Strong problem. What's that? In the fight last night. Split A and Trevor Hoffman. Finest moments, Otto. Trevor Hoffman. Michael Harris, rookie. What happened in the J. Paul D. S. fight? I'm not really sure. I don't really watch much of it.  Was it supposed to be a boxy match or a UFC fight? I don't know. I think it's supposed to be boxy. I know he pulled up in a tank though. Brad Beatty, which one? Dia. Paul? Yeah. Mitchell? Lily pulled up rolling it. Sitting on the side of the tank. And Juan Soto, blue refractor to 150. Nolan Gorman, I'm going to challenge a guy who's a mixed martial artist to a boxy match and then say I'm tougher than him. Anthony Volpe, Shay Langleyers, rookie refractor. It could have been MMA, but usually boxy. Laborator, Shohei Otani, headliners. Gunner Henderson, Buxton, and nice color coming. This is rookie design variation. Adley Rochman, gold refractor to 50. There you go. Excuse me. Nice gold rookie design variation. We'll mag that one for you. Brara. Very nice. Here at Mitchell, rookie refractor. Let's see some more big stuff out of finance here. Boxing three packs to go on this one. It's our blocker rookie. Tom Arte, green speckle to 125. And Jonathan Aranda, rookie auto. I almost said raise, but that's Tom Suzuki. Cardinals first. Yeah. There's a Cunha headliners. Bone Nailer. I was mentioning the other day, Patrick, I was not happy about the trades until I saw where the prospects of Cardinals got in return rank on their top 10 prospect list. Adley, J-Rod, Senga. I mean, how many of them are now on the Cardinals top 10 prospects? But that's not building for 2024. Max Meyer, world's finest, Mookie Betts. Bone Nailer, Adley rookie. Lenion Sosa, aqua refractor rookie. And oh, that's cool. World's finest auto, 16 of 99, Trey Turner. He had a big homer yesterday. Did he? Finally. Senga. Finally. Finally doing some of the silly. Pasquantino, you're done. Moreno, Masatake Yoshida. Yellich, Mike Trout, blue aqua vapor to 250. Ronald Acuna, refractor, Vlad Headliners. Tray Turner, purple to 250. Ohtani, and that's Rose Gold rookie, Miguel Vargas. I'm excited for a second. All red. Christopher Morell, Suzuki, Ruiz. This is rookie green refractor auto to 99. Igai Rosario. What Igai? Corby Carroll, Francisco Alvarez. All right, two packs of finest to go. Headliners, Griffey. We're about to pull a monster here for Tom. Joey Vado, and rookie auto Oscar Gonzalez. Riley Green. Here's Bowman's vest. Still lots of room if you want something ripped. Here's Tamar. And Gold Lava rookie, Jake Berger. What do you give him, Marlins? Yes. It's a 75. There's Capay and Stephen Kwan, rookie refractor. They had a sneaky of them. Sneaky, good headliners. It's Tamar and Nicoledolo, rookie refractor. That's the player that we traded for already having an impact. Abrams. And auto Luis Gonzalez. Josh Bills already claimed better than he ever was with Cleveland. Oshet Purple and Alex Thomas, rookie refractor. J-Rod, elements of excellence. Cross and James Wood. Nice. That is blue refractor auto to 150. Jet Williams, nice color match. Auto, Gumry, Christian Yellich, Atomic, Freeman. Juan Soto, Bryce Harper refractor. This freaks me out a little bit if you look at the top cam. And just down below it, like it. There we go. Like certain angles. Looks like I just have like a giant upper arm. It does look weird. Machete. Refractor auto. Arrivial on LS. Christian Pache. Blue refractor. The most boxes for one person I've done. There's Anthony Volpe. Probably 50, 60. Something like that. There's a creepy echo. Is it me or are you talking about? Talk about Jake over here. There's Jackson Holliday. I'm doing a pretty creepy echo. Jackson Holliday perspective royalty atomic. Rizzo. Auto, Estaban, or I'm sorry, Estanley Castillo. Simon Juan UFO and George Valera. All right. So 22 bum and chrome sapphire. The audio is a little muffled. Huh. Yeah, it is. We're going to be tweaking things a little bit here and there. What's up, Blizzard? Has it been doing this since I went live? How about now? Drew Baker, Cesar Prieto, and Freddie Freeman. Yellich, Faunas, Campos, and Perez. Is it about the same? All right. I'm going to mess with it after this. Sorry about that. Abrams, Rookie, Pacell, and Jordani de los Santos. Aqua Sapphire to 99. Rodolfo Castro, George Kirby, PCA, Joe Ryan, Rookie, and Jarlene Confidant. Juan Rookie, Acosta Pache, Christian Vecchero, Christian. Thanks, Vecchero. Banz Valera, Martinez, and Mejia. Come on. I see a huge hit here for Tom. O'Neal Cruz, Rookie, Garcia, Lewis, and Angelus. Scherzer, Paulino, Stanton, and Watson. Ramos, Marte, and Couser. There's Jeremy Pena, Rookie, Dominguez, Balcazar, and Ellie de la Cruz. Nice, Ellie. J-Rod, Rookie. Nice, J-Rod. Mitchell, Pujols, and Raleigh, Rookie. Roderick Arias, First Bowman Sapphire. Terry Ford, and Yesel Marobel. This is First Bowman Purple Sapphire to 25. Nice card. Danny de Andrade, Anthony Rizzo, and Rookie. Orton Sapphire to 75. Hunter Green, Mercedes, and Marte. Kira's, Trout, Vasquez, and Sheets. Yeah, I have Patrick. Yeah, it's been nice. I actually have a couple of boxes in queue of it. There's Plasencia, Mount Castle, and Trey Sweeney. Aqua Sapphire to 99. India, Grubaker, Velaide, Baez, and Weemer. All right, let's do 23 Bowman Sapphire. There's Carlos Jorge, Zach Galoff, and Carlos Jorge. That's First Bowman Orange Sapphire to 50. Churio, Gilbert, Mejia, and Gabriel Martinez. That is First Bowman Green Sapphire to 60. Jorge, Malloy, Perez, and Serda Martinez. There's Justin Crawford, Muncie, and Alonzo. James, and Lee, and Jackson Holiday. Sapphire, Otto, nice one. There you go, Tom. Nice Jackson Holiday. There's Ganal. Junior, Caminero. First Bowman Sapphire. Susak, Mesa, and Sweeney. Son, Avina, De La Cruz, and Gutierrez. Brady House, Moisespa, and Steros. And Ronnie Simon. That's Aqua Sapphire to 55. Last box of 23 Bowman Sapphires. Zavala, Holiday, Arias, and Mercedes. Thomas, Rodriguez, Juan, and Bishop. Graham, Frias, and Xavier Isaac. That's First Bowman Green Sapphire to 60. Nice. Not bad. Encarnacion, Rodriguez. Boyd, Lee, and Sanchez. And Redevis. Nice color coming. Jose, Perosa, and Ryan Clifford. That is First Bowman Red Sapphire to 10. Nice Ryan Clifford. He was just involved in the trade, wasn't he? The Met Street. He wasn't a Met Street. Yeah. For Verlander. He was with Gilbert. That's a pretty good return. Yeah, Gilbert, and him. Yeah. That's Jefferson Rojas, Sanchez, Ramiro, and First Bowmanado, Jace Avina. Osuna. Two packs to go, then we'll get to the baseball jerseys. Blider Figueroa, Michael Arroyo, Rafaela, and Vicaro. Simon, Osorio, and Colton Couser. Aqua Sapphire to 55. A louder. All right. Baseball jerseys. We have three of them. Yoan Moncada, autographed White Sox jersey. I think that's the City Jersey. That's cool, actually. That's the City Connect. Yep. Yep. City Connect jersey. Pretty good. Yeah. That's Moncada. That is, oh yeah, that is. Jose Canseco, autographed A's batting practice jersey. Hmm. That's cool. That's an interesting looking jersey. Yeah. I like the variety here. The City Connect jersey, a batting practice jersey. Nice Canseco. All right. Last box. Let's see something crazy in this one. There we go, Tom. Good luck. There you go. There he is. The Wizard, Ozzie Smith. What the, I'll get this verification out of the way. There we go. Ozzie Smith with the Hall of Fame O2 inscription. There you go. Nice Ozzie. Nice new uniform. Actually, I almost wore this today. Yep. Not signed by Ozzie. There you go, Patrick. Not Patrick. Yeah, it is. Patrick. There you go, Tom. That'll do it for your break. I mean, just a second here. We're going to recap some of the hits. Man. First bow in Sapphire Auto. Jay Savina. Sapphire Auto. Jackson Holiday. A red Sapphire to 10. Ryan Clifford. Green Sapphire to 60. Xavier Isaac. Aqua Sapphire to 55. Rani Simon. Aqua Sapphire to 99. Trey Sweeney. Orange Sapphire to 50. Carlos Jorge. Purple Sapphire to 25. Esomer O'Bell. Orange Sapphire Rookie to 75. Yeah, 75. Hunter Green. Aqua Sapphire to 99. We're done. Adela Santos. Green Sapphire to 60. Gabriel Martinez. Aqua Sapphire to 55. Colton Couser. Rookie Sapphires. Jeremy Pena. Julio Rodriguez. Steven Kwan. O'Neal Cruz. There's just some bass in there. Sorry. No, you're good. Then we had a one-of-one. Shay Gilgis Alexander. And an Adley Ruchman. That's a Top's Finest Rookie Design Variation. Gold Refractor to 50. That'll do it for your break, Tom. Appreciate it. We'll get these right out to you.


Despite recent successes with monoclonal antibody checkpoint inhibitors and disappointing results from cancer vaccine trials, I believe there is potential for patient-specific vaccines to address unmet clinical needs and improve outcomes for patients who do not respond to anti-checkpoint therapy or make immune responses to their tumours.


Hi. Welcome, everyone. We'll get started. I'm James Morris. I'm the lead organiser of the program committee. And this is now the ninth Linux Security Summit in North America, and it's been growing. I just checked with the Linux Foundation folk. There are 250 people registered for this, and we were full. We've had people for several days trying to get in on wait lists, so it's certainly grown since the humble beginnings back in Boston. So now, as everyone would know, we have three days. We've graduated from a two-day conference, and that allows us to have a more relaxed schedule. I think last year we were starting early, finishing late, and having rushing between talks. So now we have a five-minute gap in between talks for people to move, and we have good breaks and reasonable starting times. Also, we have the addition of tutorials, and while the tutorials are happening in this room, we have an unconference session on the first two days, and that is actually directly downstairs under where we are now out in the... There's like four tables out there. So with the unconference session, you just turn up and decide what you want to work on or talk about with other people, so it's for collaboration if you're not attending a tutorial. We also have bof sessions, and we'll be getting a flip chart or something like that where people can write what they want. So I know Roberto Sassu had a topic he wanted to discuss, for example, so we can write that up on the bof sessions. I won't take up any more valuable time at the moment. I'll just hand over to our first refereed presentation, David Kaplan, and Nathaniel McCallum will be talking about E-Narx, a project for secure attestation.


Clean and my wife she she messed around and called me a freelancer the other day. I said, okay, you got jokes, huh? But anyway, thank you clean. I really appreciate y'all. I thank you all like seriously I do man because so many y'all reached out and showed a lot of support So thank you. Thank you. Thank you. Thank you. Thank you. Thank you for everything that y'all have done and are doing And just showing a lot of love and support man. So I appreciate y'all now Jeremy Fowler Jeremy Fowler who has been doing a lot of cover in this whole Lamar Jackson thing From beginning to I can't even call it the end because we we get ready to get to the next step in a couple days But anyway, I'm Jeremy Fowler. He reported that the Ravens have strongly considered using the non-exclusive franchise tag Now before we get into this video before we get into this topic Again, like I said We don't know We don't know anything that's going on I know Jocina Anderson a couple days ago. She and she's been she's been on point But she reported that the Ravens they put their strongest off on the table for Lamar Jackson. All right, boom there we go and Then she also broke down how while Lamar's Camp know they they're not necessarily asking for a fully guaranteed deal But he wants more guarantees than the Sean Watson. So that would be anything over 230 mil Guaranteed because you could do a you could do a 300 million dollar contract 230 mil guaranteed you You could do a 232 million dollar contract with 230 guaranteed I mean 231 guaranteed to just be one meal that would be given but anyway, she broke that down So shout out to Jocina Anderson but back to Jeremy Fowler and really just everything like I like I said before We don't know we won't know till we know that's why any report that we see we're tripping about it We ain't chipping over that for what for what we ain't tripping over that at all So let's let's just read this one from Jeremy Fowler that was on ESPN plus He said non-exclusive tag for Lamar Jackson Several people around the league believed the Baltimore Ravens have strongly considered using the non-exclusive franchise tag on quarterback Lamar Jackson Which would deepen the intrigue on his future in a major way While the Ravens haven't officially made his decision leading up to the deadline several league executives now believe the non-exclusive tag Which allows a team to offer Jackson a contract that Baltimore can either match or decline in exchange for two first-round picks makes the most sense for Baltimore now If you use the non-exclusive tag on Lamar Jackson I think you I mean That's just you Officially sending him on his way in my opinion now I know that if they did use a non-exclusive tag the Ravens would have the opportunity to match They can match whatever deal and let's talk about how this could Benefit both sides for the Raven side for example, if somebody signs him to an office sheet Then the Ravens have some options They can decline to match whatever the offer is and they could be like all right Lamar. Thanks, but no, thanks We'll take the two first-round picks appreciate it and He would go to whatever team that is This could also Potentially help the Ravens and they could be like look see this is what the market is saying That you're worth. Look at our offer. Look at that and boom there you go And then they will it would depend on how their offer matched up with the offer that he got from the other team But again that that would sort of show the market for his services now You give up two first-round picks for Lamar Jackson Quarterback needy team. Yeah, they would even some teams that got a quarterback depending on which team it is But I know they won't but anyway That's not a bad deal Obviously you will need to sign him to a long-term extension as well, but it's really not a bad deal So this would allow Lamar to go to a team that would pay him And if he agreed to whatever terms that they came up with and the Ravens didn't match it and he would go there He would go there. Now, of course in order to go there. He would have to sign the tag But this would allow him to be like the end of our videos and be out So it's it's benefits to both sides for it now at the same time if you're the Ravens if you're Eric D'Costa and Say for instance that they want to trade him and I mean this article like saying the Ravens strongly considered using what they strongly are considering using the non-exclusive tag Well, of course, they got to strongly consider all the options So that's a no-brainer right there, but if if you're Eric D'Costa and you are really one of you you want to trade Lamar Jackson I don't think the non-exclusive tag will be the wisest move because you only give two first-round picks and that's it No more no less. That's it.  That's it If you're Eric D'Costa and you want to trade you intend on trading Lamar Jackson I don't think it will be small, but if you intend on trading Lamar Jackson Then why would you just do it for two first-round picks? That would be insane That would almost be a fireball offense for just two first-round picks like that. That'd be crazy Because you know, you can if you could intend it on trading, you know, you can get so much more than that That to me that I think that'll be bad business. It'll be bad business if they did that but It got his pros and cons I guess like we went over but we'll see what happens. Let's continue reading the article. It says Oh, I lost it. Oh, there we go. It says Baltimore has had a challenging time negotiating with Jackson So this allows him to shop his own services and bring it off into the Ravens Taking out the guesswork in the haggling over guaranteed money The tag should also fall somewhere around the 32 million far less than the projected 45 million for the exclusive franchise tag and Transferring the work to Jackson and other teams helps Baltimore maintain neutrality in any potential breakup. Okay Well, I guess there goes another plus that we didn't even think about We did talk about how yeah, it would allow another team to do the work as far as a contract So that that would be a plus for the Ravens But that last part. Oh, and then it being about 32 mil verse 45 mil Yes, that's another plus but to me the negatives outweigh the positives in my opinion on this but that last part and Transferring the work to Lamar Jackson other teams helps Baltimore maintain neutrality in any potential breakup So it would be great PR For the Baltimore Ravens even though a lot of fans will see through it But it'd be great PR for them like hey look we signed up to the non-exclusive tag and hey We we couldn't match it or we didn't match it We tried but the other teams offer was just so much better for Lamar and he just wanted to They could be like it wasn't us and they could try to pull it all on him and put it all on the other team And be like we tried because they did try they apparently been offering them contracts and whatnot But yeah, that would make the breakup a little bit smoother Nationally, but for fans, especially Ravens fans No, it wouldn't it really wouldn't but anyway We gonna see and it also said the flip side is the lower tag could damage Goodwill with Jackson at a sensitive time and the Ravens can't control any potential trade for a bigger price than two first rounders Major decision looms for general manager Eric Dacosta. So yeah, that's what we went over as far as the them getting more than two first round picks They can't get that with the non-exclusive And yeah, it just damaging doing damage To their relationship um They could because it could be like man, that's like That's how you view me non-exclusive 32 minute like really okay okay And that could really entice him to to to want to be out In my opinion, you know, I don't know what's going on in Lamar's head. None of us do but Again, these are these are just thoughts man. Um, and we we will actually get answers This week to whatever the next step is when it comes to The Ravens and Lamar Jackson and what the next move is gonna be So all these reports that we done heard all these rumors that done surface all these stories that keep popping up We getting the truth in the next couple of days team. Keep it clean. I love y'all again. I appreciate y'all a lot and we out


It was dark before I could get home to the churchyard stairs, and there, to my great trouble, I met a dead corpse of the plague in the narrow alley, just down a little pair of stairs. Samuel Pepis. Welcome to episode 012 of the SCP Foundation. Today we're going to talk about SCP-049, the plague doctor. It should be noted that this article got a rewrite yesterday, involving both the original author and DJ Cactus. We're going to talk about why the old version was problematic, and why the newer version is objectively better. Now a fair portion of this episode will contain my own opinions and thoughts about the SCP Foundation, the SCP Wiki, and the writing process, regardless. Remember that there is only one real rule in the SCP universe. There is no canon. Part one, pathology. So let's start by establishing some facts. SCP-049 was posted to the Wiki on December 10, 2009, and SCP-049 was, even prior to this rewrite, over plus 1000 in rating. Now that doesn't actually mean much, ultimately, because there are numerous bad articles on the site that are very highly rated. There's also nothing wrong with liking it in its original form, I should make that note, but I want to talk a little bit about why it has the rating it does. So if I were to sum it up in two words, I'd call it coincidence and inertia. Exact data is hard to come by for the whole history of the site, because the best tools we have for tracking voting trends can only go as far back as mid-2015, but this article is rated at just under, plus 400 at that time. It has in the last three years gained almost 700 upvotes, so part of this is certainly due to the character being included in a game called SCP Containment Breach, and the resultant popularity amongst the SCP fan community, which sometimes feeds into itself, and I'm by the way not saying that that's a bad thing. And also the character is fairly popular on YouTube channels, including, most prominently, the Vulcan, for whom SCP-049 is a regularly portrayed character. I'd also like to caution against giving too much credit to SCP Containment Breach. Now, prior to the August 2016 update of that game, the character pretty much appeared in exactly one location, and the mechanics around it were pretty bland. The character was in a place you generally needed to traverse in order to progress in the game, but SCP-049 wasn't a huge factor in the game's story or gameplay, and once you made it past the guy, that was it. So, unlike SCP-173 or 106, which are omnipresent throughout the game, or SCP-096, which was incredibly memorable, SCP-049 was largely forgettable as a character. However, in August of 2016 came the 1.3 update to SCP Containment Breach, and that all changed. In today's build of the game, SCP-049 shows up in numerous locations throughout the game, and he is designed to genuinely sneak up on you in at least one important story location. Also, the AI is more complex than approach player like it used to be. Essentially, it's important to say it's not a forgettable character today. But despite all of this, SCP-049 is almost universally understood by modern SCP writers to be bad. This can be seen in the voting data, which thankfully I grabbed before the rewrite has shifted the trends because it's changed significantly over the last day. But there were a total of 88 upvotes and 87 downvotes from wiki contributors. That gives you a net of plus one. Now as a share of total votes, that means about 8% of the articles upvotes came from people who have written for the wiki, or 8% of the total number of people who like the article have actually written an article for the site. But almost 60% of the article's downvotes come from people who have written for the site. But enough data and history and trends. Let's talk about the writing and why writers tend to not like it. Part 2. Pestilence. First of all, we're not going to do a line by line reading of the original version and point out every problem. There are some basic mechanical and structural problems with the writing, but if you want something like that there'll be a link in the description to DJ Cactus's deconstruction post. We're going to talk about why the article no longer worked from a modern writing perspective. SCP-049's original draft fell somewhere between a character based and a descriptive article. The core of it was essentially just a weird thing, but it filled out some of that with an interview log with the object itself. Like a lot of earlier site works and especially some of the more popular ones. SCP-049 acts as a blank slate and ultimately we don't really know what it looks like under the mask. We don't know what its real motivations are. We don't understand what the pestilence is, even though it seems to be a driving force for it. Now this kind of description will always allow the reader to fill in some of the blanks themselves. Further explanation is given to the object's obsession with the pestilence, though no hint is ever given as to what it is. Nothing's necessarily wrong with that because again it allows the reader to play with the concept, but we're getting now into the realms of defining nothing about the character's motivation. Here we get into the major differences between older site attitudes towards writing and today's. When the SCP Foundation was first being written, there was a big focus on creating blank slate characters that could then get fleshed out by fans in their own headcanons and by writers and tales. Literally the fourth oldest tale on the site is just a story about Able and importantly this was an incredibly vital aspect of the wiki's earliest days and part of why it was successful. But why that worked was because the circle of writers and the number of articles was remarkably low. When you have just a couple hundred characters and objects to play with, you can write almost any of them into a story and have someone else in the community recognize it, but then the community and the article list grew. Which is why about halfway through series two the wiki began to shift in a serious way towards more story and character based articles. Just creating the potential for a story was no longer enough because the wiki already had that in droves. Now the article itself needed to tell a coherent story to stand out. By the time series two came around almost every article needed to up its game and now that we're halfway through series four, a trend doesn't appear to be reversing anytime soon. All of this helps explain why by modern site standards at least, SAP 049 and a large number of the more popular older articles just don't stand up to scrutiny. And also why a fair bit of the fandom still treat the wiki as a monster manual. The more time you spend exposed to site culture the more weary you're going to get with this kind of setup that never really pays off. But from the outside you can pick out just a few objects you like and obsess over them. Perhaps you'll spend hours talking about what the pestilence really is or you'll ask yourself what set 049 on its path. There's nothing wrong with this approach. Your fun isn't wrong but neither is mine. And treating the old version of the article like it contains any answers to your questions is treating it like it's something beyond what it is. There are no definitive answers to those questions because the author never put any in the article. The author never even hinted at those answers. The lack of meat means there is no objective truth with regards to the article. And then argument you had with your friend about what the cure was involved two people with both correct and contradictory opinions. So now let's talk about how the rewrite alleviates that issue. But first are you enjoying the essay so far? Great. But here's one thing. There's a little foundation logo in the bottom right of the video. Mouse over it and then hit the subscribe button if you haven't already so you can see when I upload new content. And if you like this sort of thing hit the like button so that YouTube's algorithms pick it up and show it to more people. Alright. Thanks. And let's get back to talking about SCP-049. Part 3. The Cure. What needed to be done to update the article to modern standards was fairly simple. Just lean heavily into the character-based aspects of the article that already existed while fixing a number of the technical issues that we're not going to get into in this episode. We're not reinventing the wheel here. Switching it to a mostly narratively focused SCP would have been a poor choice. And in the hands of a lesser author we probably would have seen some sort of back story shoehorned in that betrays the article's potential. But DJ Cactus's rewrite includes two lines that really set the stage for everything that comes after them and they're pretty easy to miss so I'm going to go over them now. The first one. SCP-049 is generally cordial and cooperated with Foundation staff. That sets a baseline for what we understand SCP-049 to be. The second line is SCP-049 has expressed frustration or remorse after these killings. These very simple lines set up a baseline personality for the object and the following sections follow up on it. Now it's important to remember that in an SCP article there is going to be some necessary exposition in the description because that's what a description is. You just got to be very careful with going too far. Now in the interview logs you'll see that the baseline is expanded on. I mean SCP-049 is incredibly friendly. He sees the Foundation staff whom the reader is likely very familiar with as compatriots and co-workers. His oblivious to outside factors and single-minded focus on curing the pestilence is framed very well in juxtaposition to the Foundation staff who have their own sort of single-minded focus with regards to anomalies. So he finds a kindred spirit of sort in Dr. Ham and the interactions Ham has with SCP-049 help build up its personality. It should be noted before I continue that a rewrite like this has some very difficult limitations to overcome. One needs to stay true not just to the original concept but to a more undefinable feel of the original. And while SCP-049 is incredibly poorly characterized in the original it's not completely without character. After all there is an interview log. So with this sort of rewrite the best option is always to ask yourself could I see 049 as I've imagined say these things and do these things. Cactus its nail on the head with the rewrite and nothing that SCP-049 says or does really feels out of place. However this leads to how the character develops along the way. It's not enough to just say he kills people and feels remorse for it. You have to show the audience or they're never going to really believe it. This is why the earlier interactions with Dr. Ham are so important. The two build a rapport and you can almost imagine 049 thinks that they have a friendship. And then SCP-049 kills Dr. Ham. And 049 through a monstrous act somehow becomes more human than before. Because yes of course he's going to say that the death was necessary but he says it with a twinge of regret with just a hint of maybe I'm wrong but it doesn't quite last. Because the most human thing to do after doing something incredibly horrible is to deny that it was horrible at all. So he falls back into his earlier characterization. Pestilence has to be stopped after all. The world must be saved. This kind of back and forth of regret but justification is pretty much necessary if you're planning on portraying realistic characters that do terrible things. When we talk about the great villains very few are simply evil for the sake of being evil. The vast majority are complex characters who often do not really enjoy the things they do. Or perhaps they even wish they lived in a different world where they didn't have to do those terrible things. And that's a thing that's really very simple but in practice it's very difficult to pull off. And yet when you look at the voting trends SCP-049 has gone up almost 200 votes in the last day alone. And the net rating from site authors has increased from a net of plus one to plus 70 or so. Which would seem to indicate that Cactus's rewrite has been successful in improving the article. Part four. Recommended reading and credits. First of all all SCP wiki works used or referenced in this video are under a creative commons attribution share like 3.0 license including the following recommended works to which you will find links in the description. You should definitely read SCP-049. If you'd like to see what it looked like originally before this rewrite you can scroll down to the bottom and view the history then just hit the V on revision 58. And I mentioned this before but a lot of this video was inspired by a post from DJ Cactus on his own site. Check out both parts of his SCP-049 deconstruction post. My name is Christopher Clayton Morris though you may know me better under the pseudonym Dr. Sumerian. This podcast is licensed under Creative Commons 4.0 attribution share like unported license. Thanks for listening. And everybody knows that the plague is coming. Everybody knows that it's moving fast. Everybody knows that the naked man and woman are just shining out of fact of the past. Everybody knows by Leonard Cohen.


Thank you. While it's true, I did interview brewery owners all around the country and getting their stories that that is only fitting today for Litquake to read from a chapter on San Francisco's own anchor seam. It was started in 1896, but saved by Fritz Maytag in 1965. The name Maytag, yes, it's the same one that you find on your washing machines and for any true files in the audience same as on your Maytag blue. When his father Fred passed away in 1962, Fritz was still out in California. Growing up, he was not expected nor expecting to take over the appliance company. He explained that though his family held a fair amount, Maytag Corp was publicly owned. Quote, I'm extremely grateful nobody ever pressured me into feeling as though I had a heritage that I had to accept. When I was a boy, I never even thought of that. The dairy farm, that's a little different, I guess. Our family had this magical little business in that it interested me. Bacteria, yeast, and mold, all kinds of good stuff in there. So even from his boyhood, the blue cheese factory exposed Fritz to the chemistry of fermentation. Sitting with him, a man just entering his 70s, I could see him traveling back in time, revisiting his youth. Suddenly in front of me set momentarily young Fritz recounting experiments in his basement lab with the Baushanlam compound oil immersion microscope that his dad's friend had given him. As he grew up and through high school in Stanford, the microscope accompanied him wherever he went. Even today, he said, as if warping through time back to the present, I just get a thrilling feeling looking through a catalog of chemical apparatus. Reminding myself that he had attended university before the movie Animal House, I couldn't exactly ask about toga parties, but I still believe that college kids in the 50s partied even once who toted around microscopes. Fritz unexpectedly stated, I wasn't a great beer lover. Then he recounted a parable as jolting as if you heard your grandfather tell you that one should listen to popular music for the messages in the lyrics, or that sex is solely for procreation, and you wonder if it's merely his wholesome naivete, or if the good old days could have really been that simple. Quote, I drink beer like almost every young person does, because I didn't want to get drunk in social situations. You know, college students don't sit around drinking whiskey because you can't talk philosophy and drink a lot of whiskey. You can talk all night and drink beer if you don't drink the whole lot. I was at a loss. Then he proffered a delightful understatement or admonishment, depending on the audience. Beer is an ideal social interchange beverage. That's how he got started in his little experiment. I wanted to also jump ahead to Anchor's present. I asked if he had a brewmaster, or someone he's been grooming to take over, but I got a better story out of him. I'm the brewmaster. I have a wonderful man who's been with me since 1971, Mark Carpenter, in charge of production. I make a point of not giving up the title. Fritz explained that he feels strongly that in the small company there's a terrible danger that the manager could become uncertain about the product, lose the creativity, and the company would suffer as a result. Quote, I saw those old regional breweries where the owners, my acquaintances, knew nothing about brewing. In his rich Iowa accent, he continued, they were afraid and unable to go into the brew house and say, now look, everybody, I'm tired of this boring logger that we're trying to sell for less than Bush Bavarian. And I can't tell it apart, frankly. I want to make something interesting. I want to make something dark and hoppy, said Fritz theatrically, much to my merriment. Realizing he had given this caricature of a brewery owner a bit too much credit, he interjected, I didn't meet anyone who knew what dry hopping was, but let's just say he read about it somewhere. I silently went along with his supposition, bidding him to continue. His brewmaster would say, we can't do that. We can't have two yeasts in our brewery. It might get mixed up and ruin our wonderful flavorless logger. These guys were unable to go into the brew house and tell Otto what they wanted to do. So Fritz is in the brew house. Though he was cracking me up and the day grew colder, I knew Fritz had a million other things to do. I asked for his final thoughts, maybe what sort of legacy he felt he would leave behind. He said somberly, I'm most proud of the fact that our little brewery has succeeded by obeying the law. Most people would think I'm weird when I say that, but you see, the alcoholic beverage business is highly regulated. You could say, gee, I don't like regulation. Okay, but the law is the law. There's a stop sign over there, and if you watch it for a few hours, you'll be astonished at the way people drive nowadays by running stop signs. Fritz referred to what is known locally as the California roll, and I confessed to you, but I didn't to him that I've received that ticket. He continued, they say that's the law, but I'm so important and I'm in such a hurry, I could get away with it. And that's the way a lot of people in the beer business act. But we've tried very, very hard to follow the letter and spirit of the law. We don't do what other people do.  When Fritz first got started in the beer business, his sentiment was, gosh, we're going to make alcohol. Alcohol causes harm. It also causes a lot of good. People joke about beer guys being crooks. Could I possibly have to be a crook? It's not easy. We lose accounts every day to people who are fudging. Who'd want to lose an anchor account? I asked in shock. Oh, you'd be surprised. It's a tough world out there. Almost as surprising is that in a region imbued with free spiritedness and characterized by its granola past and silicon present, one midwesterner has remained so true to his roots.