Welcome to lecture 12 on measure and integration. If you recall, last time we looked at the extension of a measure from an algebra to the sigma algebra generated by it and slightly beyond the class of all outer measurable subsets. Today we are going to look at some special applications of this, a particular case of that extension theory for the real line and that is the topic for today's discussion namely Lebesgue measure and its properties. So, for the extension theory, we are going to apply it for the case x is equal to real line, the set is the real line, the algebra a which is algebra generated by all intervals in the real line and mu on this algebra is the length function that we have defined and we have seen that the length function on the algebra generated by all intervals is a countably additive set function. The outer measure induced by this length function which is denoted by lambda star is on all subsets of the real line and that is called the Lebesgue outer measure. So, the outer measure induced by the length function is called the Lebesgue outer measure. Let us just look at what is the Lebesgue outer measure for a subset of the real line. So, if you recall, we defined it as outer measure of a set E is look at all possible coverings of the set E by elements in the algebra, but here the algebra being algebra generated by intervals, it is finite disjoint union of intervals. So, we can write this outer Lebesgue measure as the infremium over summation lambda of the intervals i i, where the intervals i i's form a covering of the set E and these intervals are pair wise disjoint. So, lambda star of E is the infremium of the sums of the lengths of the intervals which form a covering of E and then we can take these intervals to be disjoint, because if not then we can make them disjoint. So, that is the Lebesgue outer measure for a set E. The class of all Lebesgue outer measurable sets, so lambda star measurable sets is called the sigma algebra of Lebesgue measurable sets. So, the sets which are outer measurable with respect to lambda star is called the sigma algebra of outer measurable or Lebesgue measurable sets and is denoted by L suffix R, just to indicate L for the Lebesgue and R for the real line. In case there is no confusion, we will just denote L R by simply L. So, this is the class of all Lebesgue measurable sets. If you recall, we had also defined the sigma algebra of Borel subsets of real line and that was the sigma algebra generated by all intervals and a being the algebra generated by intervals. So, the sigma algebra generated by finite disjoint union of intervals is same as the sigma algebra generated by all intervals and that is same as the definition of the Borel sigma algebra of the real line. So, these properties we had already seen. So, the length function in particular is also defined for all Borel subsets because the sigma algebra generated by A is inside the class of all outer measurable sets that is L. So, we have got that S of A that is the Borel sigma algebra is inside the class of all Lebesgue measurable sets. So, for all Borel subsets the notion of length is defined. So, this is called the Lebesgue measure. So, let us just summarize what we were saying. We were saying that the extension theory when applied to the particular case of the real line gives us the notion of length for a class of subsets of the real line which are nothing but the class of outer Lebesgue measurable sets and that includes the class of all Borel subsets. So, that is also gives us the notion of length for all Borel subsets of the real line. So, the triple R Lebesgue measurable sets the length function as extended by the extension theory this triple is called the Lebesgue measure space. So, the extension theory applied to the real line gives us the notion of the Lebesgue measure space and it extends the notion of length from intervals to the class L of all outer Lebesgue measurable sets. Let us just recall that the sets Borel subsets form a subset of the class of all Borel measurable sets is a sub class of the class of all Lebesgue measurable sets and of course Lebesgue measurable sets is a sub class of all subsets of real line. So, the question is can we say something more regarding these three classes namely Borel subsets, Lebesgue measurable sets and P R. So, let us observe which we have done during outer measures that the Lebesgue measurable sets are characterized by the Borel subsets of the real line union the null sets. So, what are the null sets? Sets in the R subsets of R such that N is contained in a Borel set of measure 0 or equivalently one can also define it as sets of outer Lebesgue measure 0. So, B R is a subset of L we know that outer measure 0 sets are also measurable. So, this and we said this class is nothing but this form the sigma algebra and that is equal to the Lebesgue measurable sets. So, that means the B R union N is equal to L. So, all null sets are part of L, but we want to characterize what is the relation between B R and L and what is the relation between L and B R. So, at present we only know that the Borel sets are subsets of all Lebesgue measurable sets which is a subset of B R. To say something more we need to look at what is called a special subset of the real line called Cantor's ternary set. So, we are going to discuss spend some time on a special subset of real line which is called Cantor's ternary set and Cantor's ternary set is an example of a set which has very nice properties and it is useful both from the topological point of view as well as measure theoretic point of view. So, let us look at what is called Cantor's ternary set ternary. It is called Cantor's ternary set because it was given by the mathematician George Cantor first defined by George Cantor and ternary set which are because it involves ternary expansion of real numbers. So, what we are going to do is it is a construction we are going to construct a Cantor's ternary set. So, as a step one let us look at the interval 0 to 1. So, I will first describe this process of Cantor's ternary set construction and then we will analyze its properties. So, what is the first step? The step is divide this into three equal parts. So, that is 1 by 3 and 2 by 3 and remove the middle open portion. So, this open portion is removed from the interval 0, 1. So, what it gives? It gives us two pieces 0, 2, 1 by 3 and from 2 by 3 to 1. So, it gives us two closed intervals. So, at the first step, at the first stage having removed the middle one-third of the closed interval 0, 1, middle one-third open interval we get these two. Now, we repeat that process again with these two subintervals. So, from each of these subintervals remove the middle one-third portion. So, that is middle one-third is 1 by 9, 2 by 9 and here the middle one-third will be equal to 7 by 9 and 8 by 9. So, this is the middle one-third portion which we are going to remove at the second stage. So, that will give us four subintervals and we continue this process. So, eventually something will be left. So, continue. So, the question is what is left? What is left is called Cantor's ternary set. So, let us analyze and let us denote this set Cantor's ternary set by the letter C. So, how do we mathematically construct this? So, that is the question. So, for that let us start with the first stage that is A0 that is the closed interval 0, 1. After having performed the first stage what is left? I write it as A1. So, that consists of two disjoint intervals 0 to 1 by 3 and 2 by 3 to 1. So, it consists of two disjoint intervals. So, let us write them as at the first stage 1 union the second one, first stage the second one. So, this portion is the first interval and this portion is the second interval. So, that is I11 and this one is I12. So, at the second stage we will be left with four disjoint closed intervals. So, let us write them as union I second stage J, J equal to 1 to 4. So, that is going to be 2 raise to power 2. And let us see what will be at the nth stage. If we continue this process at the nth stage, how many intervals will be here? So, there will be intervals, how many of them? We start with 1 at the next stage 2, at the next stage 4 and so on. So, there will be disjoint intervals J equal to 1 to 2 to the power n. There will be 2 to the power n closed sub intervals of 0, 1. Let us write them as I n J. So, these are the intervals. So, what is A n? A n is the union of those intervals which are left at the stage, at the nth stage. And what do we want? And we continue this process, we want what is C? So, how do we write mathematically C? So, the cantor set we can write it as, so the cantor set C, we can define it as intersection of A n's n equal to 1 to infinity. So, each A n is a subset of the previous one. So, let us write what is left eventually as intersection of all these A n's. So, this is what is called cantors ternary set. So, let us make some observations about this, observations about this cantors ternary set. The first observation is that the end points of the open intervals removed are in C. Say for example, 0 is not removed, is not going to be removed, 1 is not going to be removed. At the first stage, we removed the open middle one third. So, 1 by 3 is not going to be removed, 2 by 3 is not going to be removed and at the next stage, 1 by 9 will not be removed, 2 by 9 will not be removed and similarly, 1 by 3 we have already listed, then 7 by 9 will not be removed, 8 by 9 will not be removed and so on, will not be removed. So, for example, these points will not be removed. They will stay in this process of removing middle one third open interval from each sub interval at every stage. So, that means, so thus the class C, the set C is a non-empty set. It is non-empty, is non-empty. So, that is the first observation. There is something left behind and the second observation we want to show that in fact, C is uncountable, that it is an uncountable set. So, how do we prove C is uncountable? What we are going to do is, we are going to define a map from the closed interval 0, 1 to C. So, to prove this, we will define a map which is 1, 1. We will define a 1, 1 map from 0, 1 to the cantoristary set and that we prove that the cardinality of the set C is at least as much as 0, 1 and C being a subset of 0, 1 it cannot be more than that of 0, 1. So, cardinality will of C will be same as cardinality of 0, 1. That may seem a very strange observation to you that from C we have removed from the interval 0, 1. We have removed so many pieces and still what is left is as much as the points in 0, 1. So, these are properties of infinite sets. Actually, they are the characterizing properties of infinite set. The interval 0, 1 is an uncountable set and from that we are removing sub intervals and still what is left behind is as much as 0, 1. So, let us prove this fact namely that there is a 1 to 1 map for this. For this, let us start. Let us take a point x belonging to 0, 1 and consider its binary expansion. So, what is the binary expansion? The binary expansion of a point in 0, 1 is written as, so x can be written as point a 1, a 2, a 3, a n and so on, where each a n is equal to 0 or 1. So, that is the binary expansion of every point. Essentially, the idea is that the interval 0, 1 can be divided into two parts. Name first part as 0, second part as 1 and see at each stage where it lies. So, that is 0, 1, 1. And let us assume there are two different ways of writing. For some points, there are two different ways of writing binary expansions. So, we will fix one of the ways and say there is a unique binary expansion for every point in 0, 1. So, we will fix that binary expansion process. And now, what we do is the following. Construct a point y with ternary expansion b, with ternary expansion. So, y is equal to point b 1, b 2 and b n, where for every point b n, b n is nothing but 2 times a n. So, even the binary expansion, look at the nth place. Either it will be 0 or 1, double it and call that as b n. So, b n is twice as much as a n. So, each b i is either going to be 0 or it is going to be 2. So, this is the ternary expansion. So, note y belongs to 0, 1 because it is dash is dot b 1, b 2, b 3, so on. So, no integral part. So, it is going to be part of, it is a point in 0, 1. And it has in the ternary expansion, the only numbers that come are 0, 2 times a n, a n is 0 or 1, it is 0 or 2. So, in the ternary expansion of y, which is in 0, 1, only 0 or 2 appear and that implies that y belongs to c, because in the construction of the counter ternary set, we have removed the middle one-third. So, in the ternary expansion, the number 1 is not going to appear. So, each one is, so this is a part of, so this is the observation we make, that starting with a point x belonging to 0, 1 with binary expansion a 1, a 2, a n construct a point y. So, send it to the point y. So, this x is sent to the point y, which is again in 0, 1. In fact, it belongs to, so let us write more specifically, it belongs to c. So, we have got a map from 0, 1 to c and the claim is that this map, this is it is 1, 1. And that is obvious, because for every point x, we have got this binary expansion a 1, a 2, a 3, a n, the unique binary expansion. So, if you take two different points x 1 and x 2, so let us write, try to write this mathematically, that this is, so let us take a point x 1 with binary expansion a 1, 1, a 1, 2, a 1, n and so on. Let us take another point with binary, unique binary expansion, that we have fixed in the methodology, so a 2, 1, a 2, 2 and a 2, n and so on. And x 1 not equal to x 2. So, that implies, if x 1 is not equal to x 2, that implies, there exists some stage n not such that a 1, n not will not be equal to a 2, n not. And that implies that 2 times a 1, n not will not be equal to 2 times a 2, n not. And that is, this is b 1, n not and this is called that b 2, n not. That means, y 1, if we have y 1, that is point b 1, 1, b 1, 2 up to b n, n and so on. And y 2 is the other point, the image of x 2, so that is b 2, 1, b 2, 2, b 2, n and so on. Then, so if this is so, then y 1 is not equal to y 2. So, that means, this process of sending x, taking x with binary expansion is this. And constructing y with, constructing y with ternary expansion is this. So, if we send x to y, this gives us a map from 0, 1 to c, which is 1, 1 and hence, so this implies as a consequence. So, hence the cardinality of c is same as cardinality of 0, 1. And if you recall, cardinality of 0, 1 that is, so let us write this implies that c is uncountable, because 0, 1 is uncountable. Thus, c is an uncountable set. So, this is follows from our construction that c is uncountable set. In fact, let us try to now calculate. So, note that c, which is equal to intersection of a n's implies that for every n, c is a subset of a n. And what was a n? That was a disjoint union of intervals i n j, j equal to 1 to 2 to the power n. And at the nth stage, what will be the length of each where? The length of each i n j, so what is the length of the intervals, which are left at the nth stage? That is 1 over, so let us just look at the construction. At the first stage, when we removed 2, at a 1, 2 intervals were left each of length 1 by 3. So, this is 1 by 3, this is 1 by 3, this is 1 by 3 and this is 1 by 3. So, 4 intervals at the second stage of length 1 by 3. So, at the nth stage, how many, 2 n intervals of each of length, how many will be left? They are 2 to the power n intervals each of length, how much? So, here the length of each i n j, so at the second stage it is 1 by 3. First, second stage is 1 by 3 and nth stage will be 1 over, so it will be 1 over 3 raised to power 2 n minus 1. So, that will be the length of each one of them and there are 2 to the power n of them. So, what is the total length? So, sigma lambda of i n j, j equal to 1 to 2 to the power n, so that is 2 to the power n intervals each has got the same length. So, divided by 3 raised to power 2 raised to power n minus 1 and observe that this number goes to 0, it is 2 to the power n by 1 over 3 raised to power n that goes to 0 as n goes to infinity. So, that means what? That means that c can be covered by for every n by 2 to the power n intervals whose length is this and that can be made as small as they want. So, that means that the outer Lebesgue measure, so lambda star of c is equal to 0 because what is lambda star of e? It is the infremium of the sums of the intervals which cover the set c and here we have just now shown that c is contained in an which is a finite disjointed line of intervals and the total length of these intervals is becoming smaller and smaller. So, that is length of c. So, that implies that c is a lambda star null set. Hence, c belongs to is a Lebesgue measurable set and not only that c is in fact what we know is something more that if e is any subset of c, then that implies that lambda star of e also equal to 0 because lambda star is monotone and that implies that e also belongs to l. So, hence all subsets of c, power set of c is a subclass of the Lebesgue measurable sets and of course, Lebesgue measurable sets are a subset of power set of real line. Now, but c is uncountable and r is uncountable. So, what does this imply? That means this implies that l has as many elements as p of r. So, what is the meaning of this? It has as many elements as p r that is same as saying the cardinality if you know what is cardinality. Cardinality of l is same as the cardinality of the power subset of real line and if you know that the cardinality of real line which is what we call as cardinality of continuum is denoted by the small letter c. So, this is denoted by 2 to the power c. So, what does that prove? So, that proves that if you look at from the cardinality point of view, if you look at how many elements are there in the class of all Lebesgue measurable set, then it says cardinality of Lebesgue measurable sets is as much as the cardinality of all subsets. So, if you look from the cardinality point of view, you cannot say that the class of all Lebesgue measurable sets is a proper subset of the class of all subsets of the real line. So, but that does not also imply that all subsets of real line are Lebesgue measurable. So, the question still remains undecided whether the class of all Lebesgue measurable sets is a proper subclass of all, proper subclass of all subsets of real line. So, to decide this question is a bit difficult and that relates to some fundamental questions in set theory. So, let us look at, so what we have shown just now, let us just recapitulate that lambda star of c is equal to 0 and that says that power set of c is a subset of L and hence there are at least as many elements in L as 2 to the power c. So, that is the cardinality of the continuum. So, we get cardinality of L and power set is same, both have got same cardinality. So, question still remains is L a proper subset of P R. So, if you recall the answer to this question is related to some of the fundamental questions in set theory. So, if you recall we proved what is called Ulam's theorem, we did not prove it really, we mentioned what is called Ulam's theorem and I said that one can read a proof of this in the text book that we have mentioned and that is the statement of the Ulam's theorem says, assuming continuum hypothesis, Lebesgue measure cannot be extended to all subsets all of real line. There is something called continuum hypothesis in set theory, I will not explain at this stage what is continuum hypothesis because will be going slightly off stream, but it is worth mentioning here that the set theory is based on certain axioms. So, whatever modern mathematics we are doing is based on axiomatic set theory and there is a which has some kind of some axioms on which we are we are we can deal with set theory, but there is something called continuum hypothesis which relates to the subsets of real line and so on and that is not part of the axioms of set theory that is why it is called continuum hypothesis. Some people believe in continuum hypothesis and do mathematics according to that and some people do not believe in it. So, if you assume continuum hypothesis and Ulam's theorem says that you cannot extend that means not all subsets of real line are measurable. Another result which one can use which is again not part of the axiomatic set theory is the following which says that supposing you assume what is called axiom of choice. Axiom of choice is another axiom which is not part of the axiomatic set theory and one can either accept it part of set theory and do mathematics or do not accept part of it and do mathematics. So, mathematicians those who accept axiom of choice they are supposed to be doing what is called non-constructive mathematics because there are some existence theorems which assume axiom of choice helps in proving some theorems which are existential in nature. For example, proving that every vector space has a basis requires the need of using axiom of choice. You cannot prove it if you do not assume axiom of choice. There are many results in mathematics which use axiom of choice and which are not true if you do not assume axiom of choice. So, what is axiom of choice is basically saying very statistically saying given a non-empty collection of non-empty sets, you can pick up one element from each set and form a new set. So, it is how sets can be constructed when the sets are not indexed by a family which is finite in number essentially. So, it says given any indexed family of non-empty sets and that indexing set also is non-empty from each one of these sets. You can pick up one element and form a new set. So, using this one can show there exist sets in the real line which are not Lebesgue measurable. So, we will prove this result. So, assuming axiom of choice there exists non-Lebesgue measurable sets in real line. So, let us prove existence of non-measurable sets by assuming axiom of choice. So, let us start. So, what we are doing is existence of non-measurable sets. So, that is what we are discussing. So, we want to construct a subset of the real line which is not Lebesgue measurable. So, to start with consider once again the interval 0 to 1. So, consider the interval 0 to 1. So, this is the interval 0 to 1. On this I am going to define a relation so far x related to y if x minus y is a rational number. So, for x and y take two points x and y in 0, 1 and you say that they are related with each other if and only if x their difference is a rational number. So, the first observation I will just write claims 1 that this x related to y is an equivalence relation. So, what does equivalence relation mean? It means it is reflexive, symmetric and transitive. So, what is reflexive? x related to x that is obvious because x minus x is 0 and that is a rational number. And secondly if x is related to y that means x minus y is a rational number and so the difference so the negative of that that is y minus x also is a rational number. So, that implies that y is related to x. So, if x is related to y then y is related to x that is called symmetry that the relation is symmetric. And the third one is let us suppose x is related to y and y is related to z. So, x related to y means x minus y is rational and y related to z means y minus z is rational. So, if you take the difference that implies that x minus z is a rational so that implies that x is related to z also. So, it is an equivalence relation it is a reflexive, symmetric and transitive. And every equivalence relation given on a set partitions the set into equivalence classes. So, that is the basic idea that 0 1 can be partitioned into equivalence classes by this relation. So, that implies so second that implies so let us write that 0 1 can be written as a disjoint union of equivalence classes. So, let us write it as e alpha, alpha belonging to some indexing set let us call it as a, e alpha equivalence class. And recall equivalence class means e alpha intersection e beta is empty for alpha not equal to beta that is why I have written as a union with a this sign. That means equivalence classes they cover 0 1 and they are disjoint. So, there is a partition of the set on which equivalence classes are defined. .. And the third step is from each e alpha select some element x alpha and form the set call let us call it as e which is x alpha x alpha belonging to the indexing set a. So, what we are saying is using this equivalence relation partition thus interval 0 1 into equivalence classes and from each equivalence class pick up one element exactly one element x alpha select one element x alpha choose one element x alpha from each equivalence class and put them together in a box call that e and claim is that e is a set. And this is a place this is the place we are using axiom of choice. So, that means e alpha is a collection of non-empty collection of non-empty sets from each we can pick up one element and form this set this is possible only if we assume axiom of choice. So, here is the place where we are using axiom of choice. So, from each equivalence class we have picked up one element and constructed a set e. So, obviously this set e e is a subset of 0 1 because each equivalence class is a subset of 0 1 and from each we have picked up one element. So, this is a subset of 0 1.  Rationals in the interval minus 1 to 1 is a countable set. So, they can be enumerated they can be written in the form of a sequence. We are not saying r 1 is smaller than r 2 or anything. We are just giving a enumeration of the rationals. They are countable in many. So, we can write them as a sequence and construct define a set en which is e plus r n n bigger than or equal to 1. Construct a set en this. So, let us observe where is the set en? e is in 0 1 and each r n is between minus 1 to 1. So, what can you say about the set e plus r n? So, e can be 0 to 1, r n could be minus 1 to 1. So, that means each one of them is a subset of minus 1 to minus 1 to 2. At the most this sum can become minus 1 where elements of e are smaller the smallest ones 0 and possibility here is a minus 1 and the largest possible is r n is equal to 1 and e also element is 1. So, 1 plus 1 2. So, for every n en is a subset of 0 1 of minus 1 to 2. So, this implies that the union of en is also contained in minus 1 to 2. So, that is one observation. Also, if I take x belonging to 0 1, if I take an element x in 0 1 that implies x is related to x alpha for some alpha because the equivalence classes cover 0 1. So, every element x in 0 1 has to belong to one of the equivalence class. So, say it belongs to e alpha. So, that means it is related to x alpha the element that we have picked. So, that implies that x minus x alpha is a rational x minus x alpha related means the difference is a rational and so where will that rational be? x is in 0 1 x alpha is in 0 1 this is a rational is a rational in minus 1 to 1 because both could be 1 and that means that is x minus x alpha belongs to en because if it is a rational in minus 1 to 1 that must be equal to some R n and that means x is equal to x alpha plus R n and that means it is in en. So, what we are saying is for every x in 0 1 x minus x alpha belongs to is in R n. So, sorry that not that to that implies that x is equal to x alpha plus R n and that belongs to en. So, x belongs to en. So, the second observation is that 0 1 is inside the union of en's. So, that is what we have gotten. So, this construction we have got is the following that 0 1 is contained in union of e plus R n that is en n equal to 1 to infinity and that is contained in minus 1 to 2 and in this construction of the set e we have used this axiom of choice. Now, here is one observation that let us move on to observe claim that these sets e plus R n intersection e plus R n are disjoint sets for n not equal to m. To prove this, so let us take an element x which is common. So, if not x belongs to e plus R n that means x is equal to x alpha plus R n is also equal to it is also in e plus R m. So, it is also equal to some e beta plus R m and that implies that so that implies x is related to x alpha and x is related to x beta that is x is related to x element that implies either x alpha is equal to x beta if that is there should be same and that is possible implies that alpha is equal to beta. So, if alpha is not equal to beta then this is not possible. So, that says that means that these two sets are disjoint. So, this is what we have gotten. So, as a consequence let us write this as that 0 1 is contained in a disjoint union of e plus R n n equal to 1 to infinity and that is contained in minus 1 to 2. So, till now we have not done anything except we defined a equivalence relation and using axiom of choice we constructed a set e and this as this property. Now, suppose assume that e is Lebesgue measurable then there are two possibilities one Lebesgue measure of e is equal to 0 but that implies Lebesgue measure of e plus R n is equal to 0 for every n because Lebesgue measure is translation invariant and that implies that the Lebesgue measure of the union e plus R n is equal to 0 and that implies because 0 1 is inside this that means 0 1 equal to 0 which is a contradiction because Lebesgue measure of 0 to 1 is equal to 1. The second possibility is that the Lebesgue measure of e is strictly bigger than 0 then that implies Lebesgue measure of minus 1 to 2 this close interval is bigger than or equal to Lebesgue measure of this union because that is a subset of it and that is equal to sigma lambda of e plus R n and that is equal to sigma lambda of e because for every n it is same and this being a positive quantity added in infinite number of times that is equal to plus infinity which is again a contradiction because lambda of minus 1 to 2 actually is equal to 3 and 3 equal to infinity is a contradiction so either case this assumption cannot be true. So, this is a set which is in 0 1 and which is not measurable. So, what we have shown is the following that if we assume axiom of choice then there exist non-Lebesgue measurable sets in the real line without axiom of choice or without continuum hypothesis it is not known that you can construct subsets of the real line which are not measurable non-Lebesgue measurable. In fact, there is a theorem which says that the condition that assume axiom of choice actually if you put this as an axiom in set theory that every subset of every subset of the real line is Lebesgue measurable if you take that as an axiom and if your set theory axioms are already consistent then adding this new axiom to your set theory will not make any difference it will still leave it consistent. So, existence of non-measurable sets get related to fundamental questions in set theory. So, on this side we will leave it as it is saying that if you either assume continuum hypothesis or you assume axiom of choice then there exist sets which are not Lebesgue measurable. Let us turn to the other side can we say that the Borel sigma algebra the Borel subsets of real line they form a subset of for this form a subclass of Lebesgue measurable sets what is the relation between these two can we say that the Borel sets form a proper subset of the class of all Lebesgue measurable sets. So, one can show will not prove most of the things here because they are slightly technical. So, first observation is that the Borel sigma algebra of the real line which is a sigma algebra generated by all intervals is the same as the sigma algebra generated by all open intervals because one can show that every open set in real line is a countable union of open intervals actually that is using the basic topology in the real line. So, topological properties of real line come into play and not only that in fact you can take open intervals with only rational end points and if you generate the sigma algebra by them that is same as the Borel sigma algebra. So, this needs a proof so we will not prove it, but just indicate what is involved here. So, the Borel sigma algebra so this is a countable family open intervals with rational end points. So, you take a countable family of intervals and generate the sigma algebra and that is B R and one can show that the cardinality of this process of generating is exactly equal to C. So, one using these properties one shows using this construction one shows that the cardinality of the sigma algebra Borel sets is same as that of C that of the continuum and that is of the real line. Whereas, the cardinality of the Lebesgue measurable sets was 2 to the power of C so that means there exists sets. So, cardinality looking at the cardinality says that there exist sets which are Lebesgue measurable but which are not Borel sets. But actual construction of these sets is not very easy it is possible to construct such sets which are Lebesgue measurable but which are not Borel sets they are called analytical sets, analytic sets and for that we refer the our textbook for more details those who are of you are interested they should refer the textbook for more details. So, what we have shown today is that in the special case of the extension theory we get the notion of length function on a class of sets which are called Lebesgue measurable sets which include the Borel sigma algebra of subsets of the real line and the cardinality of the Lebesgue measurable sets is same as the cardinality of all subsets and if you make some assumptions like continuum hypothesis or axiom of choice you can show existence of sets which are not Lebesgue measurable. Otherwise you cannot show there is no such proof known and on the other side the Borel sigma algebra has got cardinality C which is much stricter strictly less than the cardinality of Lebesgue measurable sets. So, we will continue looking at the properties of Lebesgue measurable sets vis-a-vis open sets closed sets and the group sector on real line in the next lecture. Thank you.


Yo, Erpinion, episode number one, thanks to everybody tuning in, shout out to everybody in the building, first guest for Erpinion episode one, I've got Tony, Gem, Locke, shout out to them for coming through. What do you think Sol's going to say about what happened to Punch and the last time he was here? What do you think Sol's going to be like? Sol can't really have an angle about it. If you punch someone and didn't knock them out, your angle cannot be how I punched him in the face. You've got to try to take it as far away from him. Do you feel like the main thing that happened when Sol Punched him when Sol got banned and Caustic's name didn't really suffer? Yeah, no, nothing really did. I don't think Caustic's name really suffered. No, not at all, but quite... He made him more popular with the Donald Flop crowd. I think Jon Jon had done this thing, he battled Jon Jon had done next and I think Jon Jon, I think said, he might have said, I think one line when he's like, yo, I heard some boy knocked you out, I think.  Yeah, cause no one got knocked out, didn't they? But it's still an infamous moment, it doesn't matter if he slapped him, flicked him, whatever, he's still like... I think the angle for him is to go on about how shady it was to expose a man's family life like that. Yeah, that's where it has to go on. To be honest, if Jefferson Price had gone outside and beat Caustic over the head, I don't think he could have been mad, but Jeff didn't do that. Sol's on the punch. Come back on the sticks, Jim. If he... Yeah, so what would you do if Sol brought Jeff out? Or if Jeff brought himself out? Imagine if Jeff... Okay, if Jeff came out and even Sol didn't know Jeff was gonna come out. And I didn't know, and it was... Yeah, imagine... If Jefferson Price returned... The miracle in unison. The whole crowd, the whole ministry of sand. I think that would have been all Sol. There you go, that is an angle for him. Bring out... You feel like, yeah, it would be like the bat signal, like Don DeMarco and you sit up in New York like... What's just happening, you know what I'm saying? Scramble for his soundboard is tough, isn't it? Yeah, look, he's a good rapper. He's a pretty good rapper, man. It's a shame that he's not here anymore, but that's the thing, his legacy is that, yeah, a man just went on stage and... He's Jefferson Price. Yeah, he's Jefferson Price. When you become a verb, bro. When you become a verb. It's over, isn't it? Jefferson... I mean, there's a couple others in Don DeMarco. Well, get an EMC. That's about our term, get an EMC. I've been in three battles that my sister's been mentioned. I have nine sisters. They range from 41 to 11. Yeah. So you weigh the other six to get references. Yeah, like, it's an abortion angle. It's not 1932 no more, like women can have abortions. So one, it's a silly angle and two, it becomes real because people say it in a battle. We got said in one battle, so then they got said in the next one, so then they got said in the next one. Do you know what I mean? So then it becomes a thing. But it's just standing. I think that's a good point.  Everybody, man, the amount of chicks I know that have had abortions, it's not even... They didn't tell me and I wasn't like, oh... Yeah, that's what I'm talking about. Oh, my God! You're always like that. It's not an abortion. Round three, I was just like, I know what it cares. Like, nobody cares if you get an abortion. It was so normal. Like, it's not, you know, everybody is like... You know what's far more concerning?  Yeah, yeah, yeah. Now, there's an angle. Yeah. Because Caustic, what Caustic was most annoyed about last time is that I didn't let him hit him back, which I can understand that, because obviously my... That's fair enough. I had to... My... In the moment, my thought wasn't just like, oh, I better let him get one in. Like, obviously, in the moment, my only thought was, oh, fuck, we never used the venue before. I better stop it now. That was literally my only... Where did it happen? The cable. Onstage. Literally, onstage. Yeah, yeah, yeah. That's what I'm saying. So, I was quite... I was quite... Oh, you weren't there, were you? Yeah, yeah, literally. Soul got... Soul got up on stage and was just like, yo, what the fuck? Why did you... Why did you say that?  Obviously, Caustic went to hit him back. I kind of got him in a full Nelson or something. I probably exaggerate. I probably didn't do that at all, but... But... I did, I did. I did. I just... I had him, I was really impressed with him. Took him back. I put him... I definitely held him, and I put him in my boot. I was the barber, sleeper, holder. I had him done. I held him. I did pause, but I held him tight.   But he was obviously imagining, like, you know what I mean? Like... And he's the least aggressive guy would've ever fought, like, Soul going crazy? Yeah, I didn't expect... Of everyone in Don't Flop to go mad, I won't have picked Soul. Soul seemed like he's got that tempo. Soul seemed like... Yeah, you know what, not really stereo-typical, yeah, but he's Scottish. No, maybe Soul was going to be... Not to be stereo-typical, but most Scottish people that I know, they love a ruck. They frown on it. Like, that's just a limited hand to I know. I think it's funny that it's now officially three years ago, and when we were about to drop the announcement flying, the funniest part is, we were like, have we overestimated the amount of people that care about this battle anymore?  Like, it was... Do you know what? To be blunt, it was a win-win for us, because we were like, if people don't care about the beef anymore, both of their names would pop in regardless.  It was a great battle.  Yeah, exactly. But that was the interesting part, was just like, is this going to be something that people are going to... And people are in. And when we dropped it straight away, the entire reaction was, oh my God, literally every comment was just like, I cannot believe you set this battle up. It was fucking... It's the genuine grudge in it to have that real... It's the spirit grudge that you need it. It's everyone that takes it. How many is there left? Is there any left? Is there any envelope? Like, Yunnan is now realising that it's not about seven minutes, it's about one and a half minutes. Yes. Right, that's right. Because if you watch any unanimous battle ever, there's always good bars weaved in, but there's just a lot of stuff that don't need to be there. You know what? He's got one against Ark. Sleeping on the machine. The setup might be... Yep. Two minutes long. And that's not an even exaggeration. It may be a two-minute setup to get to your sleeping on the machine. That's a great bar. It's an incredible bar. What were you going to say in four? But you could have just put it in four lines. There was no need to do... Like, it just... Because you lose momentum and... I could actually unpick you to my head how you would have done it now as well. I can see you setting up, talking about, so you're in Osbu all the time. Yeah, you just make it... Like, just make it relevant to that punch. Yeah, just say it. And just say it and then go on to the next one. Because otherwise you can lose people in the room. As it was, the punch was so good that everyone went, ah, it went crazy over it. But you can lose people out there. And I think that's something that, yeah, like you said, a lot of the new guys...  It's because of their obsession with moties, where they do it. Because... They can ramble on and just get to the punch. If you can wrap moties, you're always impressed at how many... Relevant moties you could wear. We grew out of that when we were like 19. But to be fair, a lot of battleers aren't 19 yet, so that's fair enough. Like, it's all right, like... Because it's true, me... Thanks for that. I don't know if moties were around, I don't know, maybe, like... But like... Tony was out before the moties. Like, I'd lost to Shruffle T and Marlowe in a two-on-two long time ago, one in... But I didn't rap for them. I went in and I basically freestyleed and said to them, I didn't rap because I'm better than you. My whole angle, he was there for three rounds. I just drunkenly said, I'm too rich for you. So, like... But if the rounds they had, Jesus Christ.  Wasn't that a pamphlet? That was a pamphlet's charity thing. Yeah, yeah, like, years and years and years and... I think this shaky phone, someone's phone somewhere. They took his snatch scheme in. I think it's good, like, it's... The funny thing about Yunnan, the funny thing about Yunnan as a battler, is that when he does do well, I don't know what it is, that even though he makes so many people hate him, when he does well, you still are like, Yeah, because he's good at what he does. Yeah, yeah, yeah, when he does well, you know, that he makes these enemies, but the enemies he makes still support him when he kills it. You can feel it in a room when he battled Shuffle Man. Everybody was like, oh, Yunnan's going crazy. Like, you can hate Yunnan, but Yunnan's one of the loudest character in the dog world. Yeah, yeah. He's done, like, he's historically done annoying stuff and stuff that's winding people up and...  He's been a name in it for... Yeah, forever, forever. He was one of the earliest battles I saw with him against pH. You know what I mean? One of the earliest don't-flop battles I've ever watched was him, like, and he's just... The thing about it is that there's no one else, I don't think before or since, there's no one else who had to grow up on Yunnan camera in the fashion that he had to. You see, like, the battle-wrapped Macaulay Coco. Yeah, yeah, he's that. He just came from... You were just coming from that judge and alcohol stage, you know what I mean? Nah, he's black, he's black. Yeah, literally, yeah, like... At the time in your life... Yo, Daniel Mark Wright, I want to follow his shop of Yunnan with the teeth like that, straight away, like... But it's true, he's gone from a 16-year-old kid with an American accent to a what, 23, 4-year-old kid who's now like, all right, I know what I've got to do. I can keep that bit of rapping from 2011, this bit of punchlining from 2014 fucking... And he's still got enough legs in there to have another title shot to be on. Do you know what it is? It's great, man. You make your most mistakes, from the age of about 16 to 25.  No, like no one else has had to deal with that kind of pressure.    Chiller's methodical. He's writing. The writing level is methodical. It is good writing, but then what else is it? What else after the writing? The fact that you did, okay, the past, past, and then past, and then past, past. After you've thought of that and wrecked your brains to do that, sometimes you didn't rhyme there. That didn't rhyme my avoir. You don't rap that well. So why do you think, why do you, is that why you think past threw at him because past didn't do that? How? No, past, yeah, past is a rapper. He's a rapper. Same thing what you said, four bars set out, punch at the end. It's just... Yeah, and he knows, like, he... Both of them know what a punch is. You think Chiller dragged it out a little bit too much? Yeah. Do you know what it is to say? It's a generic. When he says, um... He knows a generic about a rapper to you. I know this one. You never heard past, what did he say? He said, um, this is better than then past flips. And then just went onto the... Then started the second round the same way with a slight difference and went, put on the... I didn't understand it. What he was trying to do. Yeah, past. Just there, no. All right, yo, Chiller, what's good, bro? Uh, what's good, bro? Man, first ever guest on our opinion, man. Straight into it, man. Seventy double you. How you feeling about the Tony Barrow? I'm excited, man. It's been a long, it's been a long time and again, the making. You know what I'm saying? We were supposed to do this earlier after I think it was the UNAN battle.  You know what, you know? We didn't, we didn't, uh, it didn't end up happening. But, uh, I'm glad it's finally going down now, man. So salute to Tony for taking a match up. I just want to make history, man.  This what happens when you battle me, bruv? You make history. You the first foreign guest on our opinion. When you soon as you battle me, history is made from. That's what happens. These are the doors that are open for people, bruv. Say, not a... Not all good history is good history. You make history and then there's infamy, man. Just don't go down and infamy, man. I've already been there, bruv. So what's the deal, man? Like, you're on a fucking, you're on a killing spree right now, man. You're on your own, man. Are you trying to add Tony to the coffin or what? I mean, I got to. You know what I'm saying? I got to. But, you know, I'm also not going to... I'm not going to sit here and discredit Tony. You know what I'm saying? I respect what he does. I'm a fan of what he does. I don't think I'm going to go into London and 3-0 Tony D. That would be silly for me to even say. You know what I'm saying? But I do think I'm going to give Tony bar for bar, pen for pen, the matchup of his life. And I'm ready for that. I look forward to that, bruv. I'm looking forward to that.  I don't think there's anybody on Earth that when I have enough preparation time, can beat me bar for bar. That's relying entirely on how much credence we give spelling. Because some of them, yeah, bar for bar. Oh, some of them is fire. But some of them, oh, A for effort, F for execution, my man. Okay. I'm just talking about, I'm pedantic like that, just reading and writing and spelling. Yeah, I don't, you Americans don't mind.  But something, it's not the same word. That ain't the same word. Don't pretend it's the same word. So don't do none of that. This is the earliest I've ever, I've got one round in the bank already for you. I've never prepped that early in my life. I've already, I'm taking you serious, bro. I've got one round in the bank already. So I don't want any stumbles. No stumbles, no slips. But that means being sober, bro. Come on, mate. All right. I'll save the drinking for after.  I want none of those. I need a clean, smooth performance all the way through. No, I'm ready.  So right at this precise moment in time, I've had like three or four battles this year. This one is the one, it goes fully global. The Chile fans get to have a look at it. So I've got a show off for them. So yeah, I'm not trying to come in there and half out anything. And I mean, I'm coming here. I'm coming to put on a classic still. I only watch pass and prep. I tried to watch bullets. You first put Jesus Christ. So why didn't you watch him versus bullets? Have you tried to watch it? I enjoyed it. I'm not even like, I was there on a day. I thought it was good. Jesus Christ. I need cocaine and Red Bull. And then I'm good to go. Yeah. Like there's no way of getting through that battle with some form of stimulants. No, to be honest, I did watch it. It was a good match. Yeah. What I think it is above all, man, it's mutual respect, man. You know what I'm saying? Like, and I know if he does, you know, it's what I do. And to me, that's what's, that's what I had. That's what I feel like I had with JC. That's what I feel like I had with daylight. Was a, not anything malicious, not any sort of real, you know, drama or anything like that. But just a competitive, like I respect what you do. I know what you bring to the table. I have to do everything I can to make sure I make every round my best round. There's one thing you can't do better than me. And that's just rap. Don't do it, don't do it. It's not even close. It's not close. Just rap where it's, just rap where it's, rap. Okay. So, so, so what you're saying to me right now? I just want to, I want to get this clear. So what you're saying to me right now is that you overall are a better rapper. Rapper. We're talking flow. We're talking everything combined. Flow, cadence, the ability to change up your flow. You don't, you don't do double time that well. You tend to rap slow over fast beats. Like, we do it different right there. We do it different. So, like, just rap where it's, like, bar for bar.  Did like the scheming and that, yeah, you scheme nice, you know what I mean? But rap where it's, but just straight up, rap where it's, ooh, yeah, yeah, yeah, yeah, you don't even close. Guy Elect and Jem, who'd you feel I might take that man? Um, so this is my thing. Whenever I predict, uh, Dialect battles, okay, he's gonna do a grime round. I heard him recently, but he doesn't usually lose grime rounds, so I always give him one round on that. But I would take, I would take Jem and I to one over on Dialect. I think Jem and I is less, not even less predictable, but I feel like he's, one, he's still growing and he's still getting better. And I feel like Dialect is kind of, he's solid, but he's passed the point of improving to me. You know what I mean? Jem and I, to me, is still growing. He's still getting better. He wraps really well. And his, his delivery on his punchlines is, is, is something different. It's unique. So I, so I think that's gonna be enough to, to get him to win for his Dialect. But that, you know, obviously I battled Dialect last year. I know that Dialect is capable of, he's still, but I, you know, I got Jem and I to one. Yo, Dialect, what's good, man? How you doing? I'm blessed, man, well done. Chillin', chillin', chillin', chillin', chillin', chillin'. I mean, we're filming this new show, man. Keep it quick, man. We're gonna keep it quick. I know you've got somewhere to be. Jem and I battle seventh birthday, like two weekends time. How you feeling, bro? I'm feeling good, you know, but like, I'm feeling good. You get me.  It's like, I'm getting into the swing of things, but I'm looking forward to it. How's it going? I need to say, preparatin', bro. How's it going? How's it going preparing? How's it been, man? I've been, I know this is one that's been on the cards for a while. How's it been? You're probably tough at first, you know, because like realistically, yeah, I rate him in it, but I rate him. So the thing is here, like, it's been a hard thing trying to maybe break down some of the things that I've, like, you know, I mean, I think it's good about him, but he's a good spit of blood, I rate him. So simple as, yeah, my strategy's at times been going on, but I've just been, you know, I mean, watching his stuff, blood and analysing blood and like, yeah, I feel like I'm in the flow of things now in it. You know what I mean? I feel like I've got the right direction. Nestle and Jem, like, where'd you rank them? Which one would you say is a better battle wrapper out of those two? Definitely Jem. I think Nestle's a better rapper, maybe possibly quick, you know what I mean? Both, to be honest, like, yeah, the best battle for me is from Jem, I'd say, but I feel a lot more like on it about him than I would for Nestle and that kind of sense of that. Oh, cool. Yeah, man. T, do your J-Short impression? Do my who impression? Do your J-Short impression? Man, I got to be around him. That's the only, I can't be around him. I got to be J-Short, man. J-Short, J-Short, man. J-Short around him. Yo, he's the dopest nigga from the UK, man. He need a title shot, bro. I promise you, man, he deserves it.  I just feel like I deserve to be the champion in the UK. I deserve a shot, bro, I deserve a shot. Like, why am I not qualified to battle Soul? You are, but you need to do a few other battles first. You're the one who gave me Jim and I. You're the one who gave me the pushover, man. That ain't my fault. And you still lost. You and you now know about him, but who lost more? That's my oldest boy, that's my oldest boy.  This is why I was angry when I battled him. You know, we good at that. This is why I was vexed when I battled him. What's good with you now, man? How do you feel about him, you now? I feel like you may go and try to be really good, man.  I feel like he's going to try to be really good.  You know what I mean? I don't know, man. I really think he... Yo, I need a performance that's scary. Like, you know what I'm saying? Like, I feel like Tony D and Shadi's performance was like, okay, now I gotta worry about that. You know what I'm saying? I really didn't see nothing about unanimous that I was scared about, except the way he looks. He's scary looking, man. The first ever battle in London. The first ever... Oh, this is my first battle in London, right? Yeah, for real.     I swear what we do, bro, you say in my first battle in London, I'm like, no, I just battled in London. No, I didn't. I never battled in London. Man, I'm a very charming young man. You know what I mean? So, you know, I gotta talk to a... You know what's crazy? Does it work for us? Like, it works for y'all? Because, oh, I love y'all acts. I was looking jealous. Like, bitch, y'all like they act same.  That's crazy. Does it work the other way around? Like, they like American acts, it's like that. No, I don't. It depends where you go, man, and what the ratchet girls will. The ratchet girls will lap your accent up. Oh, okay, then, now we go. Let me know. That's what I'm saying. You can't go wrong with the ratchet girls, the ratchet hoes. So, what do you want to do?  Is that it? After I go to the hospital from being burnt, um... I guess I battled unanimous then. You know what I mean? Like that.  That's the order. I'd rather the ratchet hoe get burnt than battled unanimous. That's my priority list. I'd rather get burnt than battled unanimous. Why'd you... What's up, man? What's up with you now, man? I'll hear you now as a vet. Like, what's up with you now in your head?  He just was weird, man. I like him as a person.  But I don't know, man. He creepy, man. He scares me, man. He look creepy, man. He creepy, man. You y'all don't feel the same way about unanimous. And I'm like, he's creepy as fuck, bro.  He didn't sit with the cool kids at the London. He didn't work with me. We know that for a fact. He didn't do that one. Yeah. He didn't, he didn't. He never hung with the cool kids. He wasn't prom cute. J Short loved Tony D, man. So I'm gonna beat Tony D for J Short. Then beat Soul. Then battle J Short. That's my plan.  And by this next year, I will be fighting for the Don't Fly title. Whoever has it in their hands. You know what I mean? And then you're gonna give it to J. Short. And then I'm battling J. Short for it and choke three rounds on purpose. Nah, man, he deserves it, man. He deserves it. I like J. Short, man. He gets a little fluck. I like J. Short. Man, he's hilarious. Have you been around J. Short? Yeah, I've been around J. Short. Man, dawg, you got to kick him with J. Short, man. Where's eternal God, man? Man, I'm real. What day am I? What day am I? You know that, man? Can I tell him? Man, in the building.  The turnip guy is in the building. He's in the building. He's in the building. Oh, man! I go to parties, man. I swear to you. If I want the party to be turned up, I pray to Kruga. Yo, man, I seen a man in the middle of Pittsburgh. He had his shirt on. Next thing you know, his shirt is tied around his head in a bandana and it's cold outside. I'm like, man, it's cold out. It's cold.  And we walked up that long-ass mountain. Remember that mountain crew? Yeah.  You don't ever go to Pittsburgh to real deal with bar hop, man. You know how you walk in mountain shoes.  No, I'm not going to try to bring y'all no half-assed performers. I'm not into that. Dad, you have to. You have to.   Nobody would care, bro. Like, no one would care, man. So, like, we're going to expect a lot of British references, or are you going to stop the Nando shit now because it's in America? Yeah, it's in America. It's like, what's the point?  What's crazy, I was always happy, like, yo, I wish it was a Nando's in America, right? But then I got one that's like, what's the point of, like, what am I going to do in the UK? You know what I'm saying? It was sad, it was kind of like, all right, I got my presents Christmas over, like. Ah, yes, I compare Nando's to Christmas. That's how much it means to me. That's real. So how did you find out about the Nando's in Chicago? Bikini. Yo, that's your favorite stuff, Bikini. In Chicago. So what about the other battles in the car? What do you feel about Shorty vs. Clips? I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. Oh, that's dangerous, man. Clips ain't coming to play though, man. I'm telling y'all, man, we on some right now.   Nah, man. Nah, we coming to show why we hit, why we are getting flown out. We going to get people their money worth. What about Tony vs. Chiller? Oh, man. I told you Tony problem, man. Tony could rap. Tony nice. I seen Tony kill a lot of people. Like, damn, like, he kill. The microwave, Blaney went off and... Shine still calling me. But yeah, like I was saying, that's Tony. I feel like, man, I don't know what he do, man.     Shit over. Er, you battling again or what? I'm trying to battle sometime soon, but I don't want to battle anyone right now. Yeah, man. You said, you said... Now you did kick it. Like after the George Brown battle, you did kind of, like, you hold on. All right, I think I know how this go now. I know how you just performance thing. I got it. After you went to your next battle, it was more kind of crazy. I heard, I mean, Er was shaking the room, man. I was like, man, I know that they Er out there shaking the room, man. Look at him. I don't know if that's a compliment or an insult. You never do. It's kind of, but, man, we never, we never all the way. This is just something about us. We never all the way be nice, man. This is always an insult with it. It's probably more you than it is me to be fair. What, an insult? Yeah. Nah, you insult everybody, man.  Why? Tell me about me. Let's go. I'm listening. Bro, you're the worst. You're the worst, man. You don't even understand, bro. I think it's because you're a rapper why you the worst, man. You, man, we're negotiating for a month. Why are we negotiating for a month? By the time we become to an agreement, the battle will be tomorrow. Like, man, I ain't even great. The battle will be the next day.  And being a difference would be like $50 or something. Like, you negotiate me a month for a $50 difference? Like, everybody heard about places like Chicago, New York, and LA. Bro, what else other than London and Amsterdam do we know? What is Amsterdam? Right. What, do you know other cities in England apart from, like, any others at all? Amsterdam. Is that, are you being serious? Yes, other than the place like Leeds and Manchester because Shadi, you mean Shadi from there, right? He's waiting, he's seeing the event. You're not, hold on, I still don't know if you're being serious or not. Amsterdam. No, no, I'm 100% serious. All right, Leeds, UK, hold on. Manchester, Amsterdam, that's all I know.  Y'all got four, I got four cities in England. He talks about brother education for you.  Like, that was in school, but, I mean, in UK, let's do a UK history class. No, never happened.  I, I, Big Nonsty, that's my, that's my uncle.  Of course. Yeah, he's the truth. Payne is hilarious, man. He laughs so funny, man. I saw the guy, he's the up, he's like, yo, he's funny as fuck, man.


Chapter 1-3 of Joel, the son of Pethuel. Hear this ye old men, and give ear all ye inhabitants of the land. Have this been in your days, or in the days of your fathers?  That which the Palmer Worm hath left, hath the Locust eaten, and that which the Locust hath left, hath the Canker Worm eaten, and that which the Canker Worm hath left, hath the Caterpillar eaten.  For a nation is come up upon my land, strong, and without number. His teeth are the teeth of a lion, and he hath the jaw teeth of a lioness. He hath laid my vine waste, and barked my fig tree. We hath made it clean, bare, and cast it away. The branches thereof are made white. Lament like a virgin, girded with sack cloth for the husband of her youth. The meal offering and the drink offering are cut off from the house of Jehovah. The priests, Jehovah's ministers, mourn. The field is laid waste, the land mourneth, for the grain is destroyed. The new wine is dried up, the oil languisheth. The confounded, oh ye husbandmen, wail, oh ye vine-dressers, for the wheat and for the barley. For the harvest of the field is perished. The vine is withered, and the fig tree languisheth. The pomegranate tree, the palm tree also, and the apple tree, even all the trees of the field are withered. For joy is withered away from the sons of men. But yourselves, with sack cloth and lament ye priests, wail ye ministers of the altar, come lie all night in sack cloth ye ministers of my God, for the meal offering and the drink offering are withholding from the house of your God. Sanctify afast, call a solemn assembly, gather the old men and all the inhabitants of the land unto the house of Jehovah, your God, and cry unto Jehovah. Alas for the day, for the day of Jehovah is at hand, and as destruction from the Almighty shall it come. Is not the food cut off before our eyes? Yea, joy and gladness from the house of our God. The seeds rot under their clods. The garners are laid desolate, the barns are broken down, for the grain is withered. How do the beasts groan? The herds of cattle are perplexed, because they have no pasture. Yea, the flocks of sheep are made desolate. O Jehovah, to thee do I cry, for the fire hath devoured the pastures of the wilderness, and the flame hath burned all the trees of the field. Yea, the beasts of the field, pamp unto thee, for the water-brooks are dried up, and the fire hath devoured the pastures of the wilderness. CHAPTER II Blow ye the trumpet in Zion, and sound an alarm in my holy mountain. Let all the inhabitants of the land tremble, for the day of Jehovah cometh, for it is nigh at hand, a day of darkness and gloominess, a day of clouds and thick darkness, as the dawn spread upon the mountains. A great people and a strong, their hath not been ever the like, neither shall be any more after them, even to the years of many generations. A fire devoured before them, and behind them a flame burneth. The land is as the garden of Eden before them, and behind them a desolate wilderness. Yea, and none hath escaped them. The appearance of them is as the appearance of horses, and as horsemen so do they run. Like the noise of chariots on the tops of the mountains do they leap, like the noise of a flame of fire that devours the stubble. As a strong people set in battle array, at their presence the peoples are in anguish, all faces are waxed pale. They run like mighty men, they climb the wall like men of war, and they march everyone on his ways, and they break not their ranks. After death one thrust another, they march everyone in his path, and they burst through the weapons, and break not off their course. They leap upon the city, they run upon the wall, they climb up into the houses. They enter in at the windows, like a thief. The earth quakeeth before them, the heavens tremble. The sun and the moon are darkened, and the stars withdraw their shining. And Jehovah uttereth his voice before his army, for his camp is very great, for he is strong that executed his word, for the day of Jehovah is great and very terrible, and who can abide it? Yet even now, saith Jehovah, turn ye unto me with all your heart, and with fasting, and with weeping, and with mourning, and wring your heart and not your garments, and turn unto Jehovah your God, for he is gracious and merciful, slow to anger, and abundant in loving kindness, and repenteth him of the evil, who knoweth whether he will not turn and repent, and leave a blessing behind him, even a meal offering, and a drink offering unto Jehovah your God.   Let the priests, the ministers of Jehovah, weep between the porch and the altar, and let them say, Spare thy people, O Jehovah, and give not thy heritage to reproach that the nations should rule over them. Wherefore should they say among the peoples, Where is their God? Zion was Jehovah jealous for his land, and had pity on his people, and Jehovah answered and said unto his people, Behold, I will send you grain, and new wine and oil, and ye shall be satisfied therewith, and I will no more make you a reproach among the nations. But I will remove far off from you the northern army, and will drive it into a land barren and desolate, its four-part into the eastern sea, and its hinder-part into the western sea, and its stench shall come up, and its ill-saver shall come up, because it hath done great things. Fear not, O land, be glad in rejoice, for Jehovah hath done great things. Be not afraid, ye beasts of the field, for the pastures of the wilderness do spring, for the tree beareth its fruit, the fig tree, and the vine do yield their strength. Be glad then, ye children of Zion, and rejoice in Jehovah your God, for he giveth you the former reign in just measure, and he causeth to come down for you the reign, the former reign, and the latter reign in the first month, and the floors shall be full of wheat, and the vats shall overflow with new wine and oil, and I will restore to you the years that the locust hath eaten, the canker worm and the caterpillar, and the palmer worm, my great army which I sent among you, and ye shall eat in plenty, and be satisfied, and shall praise the name of Jehovah your God, that hath dealt wondrously with you, and my people shall never be put to shame, and ye shall know that I am in the midst of Israel, and that I am Jehovah your God, and there is none else, and my people shall never be put to shame, and it shall come to pass afterward, that I will pour out my spirit upon all flesh, and your sons and your daughters shall prophesy, your old men shall dream dreams, your young men shall see visions, and also upon the servants and upon the handmaids in those days will I pour out my spirit, and I will show wonders in the heavens and in the earth, blood and fire and pillars of smoke, the sun shall be turned into darkness, and the moon into blood before the great and terrible day of Jehovah cometh, and it shall come to pass, that whosoever shall call on the name of Jehovah shall be delivered, for in Mount Zion and in Jerusalem there shall be those that escape, as Jehovah hath said, and among the remnant those whom Jehovah doth call. CHAPTER III For behold, in those days and in that time when I shall bring back the captivity of Judah and Jerusalem, I will gather all nations and will bring them down into the valley of Jehoshaphat, and I will execute judgment upon them there for my people and for my heritage Israel, whom they have scattered among the nations, and they have parted my land, and have cast lots for my people, and have given a boy for a harlot, and sold a girl for wine that they may drink. YAY! And what are ye to me, O Tyre and Sidon, and all the regions of Philistia? Will ye render me a recompense, and if ye recompense me, swiftly and speedily will I return your recompense upon your own head? For as much as ye have taken my silver and my gold, and have carried into your temples my goodly precious things, and have sold the children of Judah and the children of Jerusalem unto the sons of the Grecians, that ye may remove them far from their border. I will stir them up out of the place whither ye have sold them, and will return your recompense upon your own head, and I will sell your sons and your daughters into the hand of the children of Judah, and they shall sell them to the men of Sheba, to a nation far off, for Jehovah hath spoken it.   Let the nations bestir themselves, and come up to the valley of Jehoshaphat, for there will I sit to judge all the nations round about. Let ye in the sickle, for the harvest is ripe. Come, tread ye, for the wine press is full, the vats overflow, for their wickedness is great. Multitudes, multitudes in the valley of decision, for the day of Jehovah is near in the valley of decision. The sun and the moon are darkened, and the stars withdraw their shining, and Jehovah will roar from Zion, and utter his voice from Jerusalem, and the heavens and the earth shall shake. But Jehovah will be a refuge unto his people, and a stronghold to the children of Israel. So shall ye know that I am Jehovah your God, dwelling in Zion, my holy mountain. Then shall Jerusalem be holy, and there shall no strangers pass through her any more. And it shall come to pass in that day, that the mountains shall drop down sweet wine, and the hills shall flow with milk, and all the brooks of Judah shall flow with waters. And a fountain shall come forth from the house of Jehovah, and shall water the valley of Shateem. Egypt shall be a desolation, and Edom shall be a desolate wilderness, for the violence done to the children of Judah, because they have shed innocent blood in their land. But Judah shall abide forever, and Jerusalem from generation to generation, and I will cleanse their blood that I have not cleansed, for Jehovah dwelleth in Zion.


This video explains how to include newspapers, magazine or newsletter articles without volume or issue numbers in your reference list, according to APA Style 6th Edition. I will show you how to reference both print and online articles, including what to do if there's no author or no page numbers given. Most journal and magazine articles will have volume and issue numbers. In that case, please watch our video, How to Reference Journal Articles. Let's get started with how to reference a print, magazine or newsletter article without volume or issue numbers. Normally, in parentheses, you only include the year of publication. But if there's no volume or issue number, also include the month, or the season, or the day it was published. Remember, in APA Style, the first letter of every word of an article title is not capitalized. But the first letter of every major word in a magazine's title is capitalized. The magazine title is also in italics. Newspaper articles are cited the same way, with the year, month and day in parentheses. The first letter of every major word in the name of a newspaper is capitalized and in italics. The one big difference is that when you are citing a newspaper article, you use the abbreviation p. to indicate the page number. This is the only time in your APA Style reference list that you ever do this. Often with newspaper, magazine and newsletter articles, there is no author listed or the pages are unnumbered. In that case, start with the title of the article instead of an author's name, followed by the date. If the pages aren't numbered, use the abbreviation n.pag. to indicate no page numbers. For online sources, APA recommends providing the DOI or Digital Object Identifier. Most online newspapers, magazines and newsletter articles don't have a DOI number. In this case, APA recommends you provide the URL for the home page. Don't just copy and paste a direct link to the article. This is one time when a librarian will actually tell you to use Google. Google the name of the publication. It's helpful to use quotation marks around it. This is the URL you need to copy and paste. After the page numbers, type Retrieved By and paste the URL of the magazine or newspaper's home page. The URL should not be hyperlinked or underlined. It should be in the same font as the rest of the reference, with no period after it. Also, for online newspapers and magazines, you do not include any page numbers in the reference list, even if they are available. Nor do you need to use the abbreviation n.pag. For in-text citations, check out our video Referencing Sources in APA Style, a basic introduction. Or visit the Munn Libraries webpage for our full online APA Style Guide and to chat live with library staff.


Welcome to this Farm Accounting 101 presentation for QuickBooks for Farmers. Today we will be covering Lesson 3, Introduction and Sample Entries Number 5 and 6. I'm Robert Page, CPA and Regional Extension Agent with the Farm and Agribusiness Management Team with the Alabama Cooperative Extension System at Auburn University. The ACE's Farm Analysis Program worked with Alabama farmers for over 20 years, helping them to develop and maintain accurate financial and production records. In developing this series, we reviewed the different types of checks and deposits made in farm record keeping by these farmer clients and eventually selected 10 very typical farm accounting entries. We believe if you can learn these 10 sample entries, you will find the majority of your accounting entries on your farm to be similar. In today's lesson, we will be reviewing the next two entries. Recording a Farm Debit Card Transaction for Sunday lunch and Record Farm Supplies Purchase with a Credit Card. The other entries will be covered in other lessons. Lesson 6 will cover useful financial and urine closing notes that the farmers should find useful. As a reminder, these 10 accounting entries are the same entries used in the ACE's Excel for Farmers YouTube presentations. For those viewers who prefer using an Excel spreadsheet to keep farm records, please visit the ACE's YouTube channel for Farm Accounting 101 Excel for Farmers. Caution, these lessons do assume the viewer has some familiarity with QuickBooks. Our objective is to show how these sample transactions can be entered into QuickBooks quickly and accurately to produce useful farm financial and production records. In other words, this is not an introductory course on QuickBooks. Intuit and other vendors offer a wide array of training videos and manuals on this excellent software. As we have said in previous lessons, the Farm Accounting 101 series is intended to help Alabama producers to improve their farm financial literacy. In this QuickBooks for Farmers series, we are expecting three different types of viewers to be watching. There are farmers or other agricultural producers, farm family members who are untrained in accounting, and management accountants who are trained in both accounting and business record keeping who may or may not be part of the farm family. This conclusion and assumptions presented in today's presentation are based on my years of experience working with farmers, their families, and their accountant tax preparers in the ACE's Farm Analysis Program, as well as my time teaching ACE's financial record keeping small classes, as well as my time in public accounting prior to joining Extension. If we're going to be discussing farm accounting, we need a sample farm QuickBooks file. So we created the ACE's poultry farm number one. The key features of this sample farm include over 500 sample checks, and other transactions posted for this farm for the 2020 crop year. An expanded chart of accounts based on the standard farm analysis client file setup. Use of the class option to classify income expenses to one of three farm enterprises, which include poultry, livestock, and road crop for this sample farm. This class feature really can help us see which farm enterprises make the most money for the farmer and which enterprises need to be reviewed for possible changes. We will see some sample reports using this class option shortly. Now let's move on to sample entry number five. What are the facts? In this example, we're using the farm checking debit card to pay $42.40 for Sunday lunch for the family. This is a family living non-farm other expense account. In this example, the farmer is using farm funds, which are business funds to pay for a personal non-farm business expense. Therefore, this expense should be posted to either a non-farm expense account or an equity or net worth withdrawal account. Using standard farm analysis accounting methodology, this expense is charged to family living, an other expense account. Now let's change this view to look at QuickBooks. We're going to enter the check. So we write the check. Today's date. There's the check number. And let's just assume this is a local restaurant. Let's see what we have down here. We've got a restaurant. Well, we don't have a restaurant, so we'll just use local shop. And so we're going to pay $42.40 for lunch. And this is Sunday lunch with the family. Before I go any farther, let me point out I made a mistake. This is not a check number 2147. A debit card transactions would be either a electronic funds transfer or a DEB for a debit. You come up with your own code, but you need something other than a check number to identify that this is a non-check item. Now, normally we might charge this to a business expense of some sort, but this is not a farm expense. So we can't charge this to livestock surprise or miscellaneous farm expense like meals and entertainment. It's not meals and entertainment because that is non-farm business. This is taking the family to Sunday lunch. So we have to scroll all the way down in our chart of accounts down to other income. We've got in this particular farm, we've got some rental houses. So non-farm business, non-farm business, we've got two rental houses for this. And so now we move into the family living section of under other expenses. So in the standard farm analysis chart of accounts, we could have just one family living expense and just have it as 074 family living. This is the standard farm analysis numbering system. So contributions, medical expense, life insurance, cash withdrawals, food, clothing, personal items, entertainment, recreation, transportation, you know, could be any one of these. But we'll make it simple and just simply charge it to family living as an other expense item. Sunday lunch with family at local restaurant. Since this is non-farm business, you know, it's not going to be in here. It's not livestock, roadtop, grow crop. It has nothing to do with farming. So we're going to save and close. Remember, this is a family living other income or other expense account. That's where it is. So we're going to do a save and new. Nope, no class. Now let's take a look how that appears. Here's our aces poultry farm for the month of January 2021. And if you take a look so far of what we've entered in the previous lessons we've got, this is a class report. And so we've got some fertilizer crop. Here's this area up here is farm income expense. This area down here is other income and expense. And there's the sale of assets we just covered a few minutes ago. And here we are in other expense called 074 family living for $42.40, as well as my other favorite. If you don't know where to put something, just put it in 999 as my accountant. So if I wanted to see what this was, I could put it over this $42 and just double click. And up comes the check information and I could double click again. And there's the check that I just entered. Very simple, very easy to find out what it is. And this is a classic example so far of the profit and loss report by enterprise. We've got some livestock information here, some row crop information, but nothing yet on poultry. So that completes that particular entry. So now let's go back and take a look at it. As an accounting entry. So the accounting entry for the family dinner is family living, which is another expense account is a debit for $42.40. The offsetting entry is cash farm checking, which is a current asset, which is credited for $42.40. And total debits and credits are $42.40. Our next entry is sample entry number six, purchasing farm supplies with a credit card. What are the facts? Well, in this example, we will assume that the farmer used the ACES poultry farm Visa credit card, which is reserved for farm operations to buy $227.80 of cattle feed for their livestock operation. This purchase may be the only purchase on the card for the month or may be one of several purchases made by the farmer on this same card. This will be our first use of the enter credit card charges screen on the QuickBooks home page. So we're going to transfer over to QuickBooks. Here's our home page. And if you take a look right over here in the right hand corner, it says enter credit card charges. Now, if you take a look at credit card right up here at the top, we have built an account number called 821-431 Farm Credit Card. And it is identified in the chart of accounts as a credit card. So our purchase is to the local farm supply. So Farm Supply Store. We use today's date. If we'd have an invoice number would be 1234. And the dollar amount is $227.80 purchase of purchase of 20 bags of 50 pound cattle feed. So we're going to charge that scroll through there and we'll be finding livestock expenses right there. So livestock supplies. And once again, we can copy and paste. Now notice right here at the top, we've got attached file. So if we've got a copy of the invoice from the Farm Supply Store along with a credit card's receipt, we can scan both of them and attach it to this particular screen right here. Very, very important to be able to document whatever your accounting entries are. And if you have a scanner, it'll become your best friend. Just scan in the receipt you've got, attach it to this document and it's there. All right, now this one is going to be charged to livestock operations because this is cattle feed. That completes the entry. Save and close. Now let's go back to PowerPoint and take a look at the accounting entry. This is what that accounting entry looks like for this particular credit card transaction. Feed, which is expense, is at expense. Charged to 227.80 as a debit. Now in the QuickBooks entry I just made, I charged to livestock supplies. I could have charged it to feed. Either one is fine depending on the way you keep the books for your farm. The offsetting entry is for farm credit card payable, which is a current liability for $227.80. Total debits and credits, 227.80. Here's a big note. This is one purchase on the farm credit card for the month. There may be several more purchases during the month on this credit card. As part of the month in routine, the credit card statement will have to be matched up to the QuickBooks credit card entries to make sure the ending dollar balance on the credit card statement matches the ending dollar balance for the farm credit card payable account in QuickBooks. That's just very important that you may have 10 transactions on this credit card. You need to have all 10 of them posted in QuickBooks so that when you're reconciling it, the dollar amount in QuickBooks matches the dollar amount on the credit card statement. Now just as a point of reference before we get away here, since we've got two different, we've got a feed account. We've got a livestock supply account. Let me show you how easy it is to find and correct checks. We go back, check register, write checks or check register, either one. So we'll just use check register. It's not going to be in the check register. It's going to be in credit card charges. And we'll go find it right here under livestock supplies. And if we want to change this to feed, there it is, crop or feed purchases. That shows you how easy we can make a correction. And it happens all the time, friends. We just need to be able to change and correct it on demand, save and close. Yes. Now we'll go back and that completes our lesson for today. Lesson three for QuickBooks for Farmers is part of the ACES Farm Accounting 101 series and has been produced by the Alabama Cooperative Extension System Farm and Agribusiness Management Team at Auburn University in Auburn, Alabama. For additional information on the Farm and Agribusiness Management Team and other ACES programs, please visit our website at www.aces.edu.


Further, talking about privacy and confidentiality, we need to see the importance of disclosure behavior. The disclosures are very important because psychologists may disclose confidential information with appropriate consent of an organizational client, the individual client, patient or any other legally authorized person on behalf of the client, patient, unless pro-hepted by the law. So, until and unless the law prohibits them, they could disclose centered information. This is very much important to look especially in current scenario where we see that psychopathological behaviors are getting increased in number day by day. We see a lot of therapeutic centers where multiple people come for counseling and guidance. And some of them decide that there are a lot of specific traits in them, maybe they can cause potential violence to other people, they can probably cause harm to other people and they can abuse people because psychological traits are evident in them, from their verbatim, their actions, their verbal and non-verbal behaviors. In this situation, when these things are being highlighted, it is very important for those centers to disclose these things to the family, to relevant people, to law enforcement agencies so that they can properly rehabilitate those people. Each individual comes to a therapy session and repeatedly says that they have these thoughts that they have set a fire to a building. It is very important to repeatedly say that they have these thoughts that they have set a fire to a building. Or they say that they have suicidal thoughts, or they are thinking that they steal something from their house and sell it. If there are any such thoughts, then at that stage, disclosures are very important. Because if you do not disclosure at that stage, then maybe that individual can harm someone, can murder someone, can cause harm to some place, can become a threat to the state. So in such situations, disclosures become very important. So you know, you see, the students of psychology need to understand that this is such a sensitive profession. And in this, properly gaining knowledge, understanding the laws, applying the API code of conduct and understanding all the nitty-gritties is very important. Now, psychologists disclose confidential information without the consent of individuals only, as mandated by the law, we are permitted by the law for a valid purpose. That is, where there is a valid purpose, you can disclose it without telling that individual. Provided needed professional services and obtain appropriate professional consults. Where it has been confirmed that there can be any kind of danger because of it. And after consultation, you decided that yes, it is being diagnosed that he can do this kind of act or his impulse control is very severe and we need to manage it so that we can help the external. So it is very important to disclose it there. Protect the client and patient psychologist or others from harm and in any kind of harm it can be psychological, it can be financial, it can be economical, it can be physical. If it is in his mind that I will damage someone's social repute even then, this is a situation which is causing harm. Obtain payments for services from a client and patient in which instance disclosure is limited to the minimum that is necessary to achieve the purpose. Now if you are working with someone on payment, his therapy, his treatment, his assessment is done and there is some kind of a trade. So as much information is necessary, you have to reveal as much as you want. It is not that you will public all his background. Right? It is not that you will tell that how many times he has done things like this before and what is his history and past. But the current scenario, his disclosure will be relevant or important. Then another important thing is consultation. When consulting with colleagues, confidential information can't be disclosed that reasonably could lead to the identification of the client with whom they have a confidential relationship unless they have the consent of that person. For example, when we play a game called Kasati, we ask different questions and an individual reaches to that specific personality while assessing it. Similarly, if I am telling from my side that I am giving confidential information but I am not even mentioning the name. But I am saying that such working women who are professionally very recognized and reputed, they work in an engineering university and they are the only women who have become professors. So you would be able to identify the first women engineer professor in Pakistan. For that, I don't need to tell the name. Similarly, if I am talking about another profession where there is a prominent individual or who has prominent features, if I am talking about anyone without taking their name, anyone could judge which family I am talking about or which noble person. So in such a situation, to reveal people's identity, we should avoid such probes or comments so that the general public can reach that individual. Only information necessary to achieve the purpose of consultation can be disclosed as I just mentioned. Also, confidential details for didactic or other purposes. Psychologists cannot disclose in their writing and lectures confidential, personally identifiable information. You cannot easily identify that information in your writing and you cannot bring it into any discussion and you cannot bring it into any case report. As I just mentioned that you are writing this way but the particulars are so prominent that anyone could assess who you are talking about. Then they take reasonable steps to disguise person or the organization. It is very important to take care of the sanity of the organization and take care of a particular individual's sanity. Person or organization has consented in writing until and unless that particular person or that particular organization has consented that you can write our name. And in very rare cases it happens. Generally people are not open in giving permission to any such academic or any such professional activities. Then they have legal authorization. If someone is defamed like this after telling them that this individual is taking treatment for this thing and these are the issues then they have the legal right to sue that psychologist. So this point should also be taken into consideration that we should not get involved in any such act after which we have to see its legal consequences. So to breach confidential information, to protect confidentiality, to understand its limits and to see its levels and limits in generalizability is very much important aspects in human relationships.


The G7 leader seeking to rival China have adopted the plan to support lower and middle income countries and build a better infrastructure. President Joe Biden said he wanted the US-backed, build-back better world plan to be a higher quality alternative to a similar Chinese program. China's Belt and Road Initiative BRI has helped finance trains, roads and ports in many countries, but it has been criticized for subtling some with debt. In a statement at the summit in the English county of Cornwall, the G7 leader said they would offer a values-driven, high-standard and transparent partnership.