So this is the dissection of the gluteal region, this muscle that we see in front of us, this is the gluteus maximus, it's the largest muscle of the gluteal region in the superficial compartment. So it takes origin from the coccyx of the sacrum and from the ilium behind the superior posterior gluteal type and all the fibers they converge. Three-fourths of the fibers get inserted onto this tough structure that we see here. This is the iliotibia tract. Iliotibia tract is the poneuronic expansion of the insertion of the three-fourths of the gluteal muscles, the gluteus maximus and this goes down and gets inserted onto the tibia in a tubule called the adduilateral tubule of the gyr. The deep one-fourth of the gluteus maximus, which we can see here, gets inserted onto a tubule on the posterior surface of the femur or the gluteal tuberosity. So this is the origin and the insertion of the gluteus maximus. The nerve supply is the inferior gluteal nerve, which I should show you just now. The most important action of the gluteus maximus is extension of the hip and lateral rotation by virtue of its directional fibers and excretion. Now we will reflect the gluteus maximus but before that I need to say one more point. The gluteus maximus is partly a posterior muscle and partly an active muscle because it has got both type 1 and type 2 fibers. So it is responsible for sitting up, standing, climbing stairs apart from its other actions. Now I have split the gluteus maximus here and I am reflecting it. It was receiving numerous blood vessels, all of which were coming and all of which have been cut and we can see this is a branch from the inferior gluteal nerve which is supplying the gluteus maximus. Under the gluteus maximus, between the gluteus maximus and the ischial tuberosity there was a bursa and we can see the remnant of the bursa. This is the ischial bursa. In normal circumstances there is also a bursa between the gluteus maximus and the greater bursa. Finally there is also a bursa between the and that is known as the gluteus femoral bursa. These are the three most important bursa. Now we have reflected the gluteus maximus of the nerve but I am going to show you the nerve just a little later. The next muscle that we see in front of us is this one. This is the gluteus medius. This takes attachment from the posterior between the posterior and the anterior gluteal line of the ilium on the outer surface and the fibers almost descend on vertically as you can see here and they get inserted onto the greater trochanter. So this vertical descent of the fibers is very significant. This acts as a stabilizer of the pelvis especially when standing on one leg. And more importantly when bursa stands on one leg the opposite side pelvis wants to sag now. So therefore when these fibers contract they keep the pelvis stable and stable. So this is a very important action of the gluteus medius. The important action of the gluteus medius is abduction and medial rotation of the hip. Now we have cut open the gluteus medius here also and I am going to reflect. So this is the gluteus medius. I will show you the nerve supply just now. And once we reflect the gluteus medius we can see here another muscle on the left side and that muscle that we see here that is the gluteus minimus. The gluteus minimus also takes origin from the outer surface of the ilium between the anterior and inferior gluteal lines and the fibers also descend down almost vertically and they also get inserted onto the greater trochanter deep to the gluteus medius and they also have the same action as the gluteus medius namely abduction and medial rotation of the hip. But it is less important than the gluteus medius. And finally we have in the superficial groove a muscle which is between my finger here within the iliotibial tract and a little bit of that is visible here. Can you bring the camera? This is the tenserfation which takes origin from the outer surface of the iliac crest and it is a triangular shaped muscle something like this and the fibers then converge onto the iliotibial tract and little bit of the iliotibial tract have removed to show the muscle here and then the fibers. So if we were to look from behind we will see that the gluteus maximus posteriori and laterally the tenserfation both of them get inserted onto the iliotibial tract and the iliotibial tract is the aponeurotic expansion which goes up to the tube of the tibial. The iliotibial there is a feature that also has the same action in the abduction and medial rotation of the hip. And it is also supplied by the superior gluteus nerve just like the gluteus medius and the gluteus minimus. I will show the nerve just a little bit. So gluteus maximus has been reflected gluteus medius has been reflected and this is the gluteus minimus. So these are the superficial gluteus. Now let's take a look at the deep gluteus muscles. To understand the deep gluteus muscles let's start the first one here. This muscle that we see here in front of us is a pear shaped muscle. This is the biophonist and I have lifted up the biophonist. This is the biophonist and if I put my finger inside the, my finger is going inside the pelvis. It has gone inside the pelvis because the biophonist takes origin from the inner surface of the sacrum and it comes out. And it is coming out to this opening here where my finger is gone in. This is the greater shattered pyromania. So it comes out through the greater shattered pyromania and when it comes out it divides the greater shattered pyromania into a suprapyriform compartment and then the fibres get inserted onto the narrow tendon to the tip of the greater pyromania. So this is the pyriformis. The next muscle of the deep gluteus. Let's go straight down. This is the sciatic nerve which I'll describe just now that we retract the sciatic nerve. Can somebody hold the sciatic nerve for me on that side? Somebody who is wearing sciatic nerve is. You can use this instrument to track between the folds. We see this muscle here in the sacrum. This is the quadratus femoris. The quadratus femoris is the lowest of the deep gluteus muscles. It takes origin from the Istio tuberosity and it's a quadrangular shaped muscle that's why we call it the quadratus femoris and it gets inserted onto the quadrate tubercle on the posterior surface of the femur. That's why it's called the quadratus femoris. I'll tell you the nerve supply just now and the actions just now because they're all together. If we look above the quadratus femoris we see yet one more muscle here. This is the inferior gemulus. This muscle. It's a very small and a very friable muscle which takes attachment also from the Istio tuberosity and it goes and gets inserted and above that we see yet one more muscle. This one. This is the superior gemulus which takes attachment from the Istio spine and gets inserted. The superior gemulus and the inferior gemulus both of them get inserted onto this tendon that I have picked up here. This tendon. This is the operator internus tendon. So let's take a look at the operator internus muscle. For that I will draw your attention to this surface. This is a region called the Istio anal muscle and my finger is on the operator internus muscle. This is the operator internus muscle. The operator internus muscle is on the inner surface of the femoris and on the inner surface of the operator membrane. So if you look at my finger, the attachment of the operator internus is in this plane. The tendon of the operator internus it makes a 90 degrees bend in this direction. So this is the axis of the tendon and this is the axis of the muscle. Then you see the tendon is making a 90 degrees bend and the tendon then goes through this passage where my instrument is going and this is the lesser shining forever. So the operator internus muscle takes origin from inside the pelvis and the tendon makes a 90 degrees bend around the Istio spine and it then travels like this and in the region of this of the Istio spine there is a bursa or the operator internus bursa because there is friction between the tendon and the Istio spine and this is the axis. Muscle tendon 90 degrees bend around the Istio spine which is just under my next finger and this is that same operator internus tendon which I mentioned just now. This white structure and this gets inserted all through the inner surface of the greater trochanter called the trochanteric bursa. The action of all these deep muscles piriformis superior gemulus operator internus inferior gemulus quadratus femoris action is lateral rotation of the femur lateral rotation of the hip joint. The piriformis is supplied by the nerve to piriformis which comes out from inside the pelvis therefore we cannot see it in this dissection. The superior gemulus and the operator internus they receive the nerve supply from the same nerve namely the nerve to operator internus. The inferior gemulus and the quadratus femoris receive the nerve supply from same namely the nerve to quadratus femoris. All these nerves are coming out from inside the pelvis from the sciatic nerve therefore we cannot see them here. So, this is about the deep group of muscles.


It all started on May 13th when I was detected with a few different cycles. I was immediately admitted in the emergency ward in the Merlimpal hospital. The officials, nurses and the doctors were very kind, caring and patient. The registration was fast and then I was shifted to a semi-deluxe ward where I met my doctor, Dr. Vinod Reddy and my three nurses. I was not allowed to have any fluids nor any food and I was administered with u-fluids and glucose. I had a comfortable bed in my ward and I was with my constant companion, my TV. I was also ready for my big day. It was the day of my surgery. I was taken inside the operation theatre about one o'clock. When I went inside the OT, I felt like a warrior. The doctors did not want me to panic and they were very encouraging. So they were kind enough to answer my questions in a very simple manner. After administering anesthesia, it was like one and a half hour nap. After that, my surgery was over. Many of us think that surgery will give us bad thoughts and will always be negative. But I didn't take it that way. I felt it was like a one-time experience. So I went through it positively and the Merlimpal officials helped me do it. Right now we are in the coronavirus pandemic. Doctors, nurses and health line workers are acting as the frontline warriors. I would like to thank them wholeheartedly. Thank you so much everyone.


What is going on everybody is Stas here welcome back to another video So in this video, we're going to be doing an overall market update looking at the Dow Jones the S&P 500 and the Nasdaq We're going to be talking about one trade that I made today on the 4th of February in 2019 As well as briefly taking a look at what Google reported in terms of their earnings So I'm sure all you guys saw in the title that the tech stocks have continued their run And if you guys been paying attention to the stock market over the past week and a half You saw that the tech stocks, especially Facebook and Apple have been on an absolute tear And that was nothing different than today guys today We saw much more of that with Apple being up $4.75 per share Facebook up $3.50 per share We saw Amazon up around $7 per share today Microsoft up a solid $3 and of course Google just reported their earnings and they're up around $22 at the close We'll take a look at the stock in a little bit to see how it's reacting to their earnings after we talk about the earnings So stay tuned in a couple of minutes later on in this video We're going to be talking about that So now that we saw a little bit on what the tech stocks did And of course we're going to take a look at the charts in a couple of minutes Let's take a look at what the SPX the Dow and the Nasdaq did So we can get an overall idea of how the market pushed today We'll take a look at some different time frames and let's hop right into it guys So the SPX the S&P 500 the 500 largest companies in the United States of America Up around $18 today up 0.68% on the close We can see here guys like we've been knowing oops not that close Like we've been knowing over the past couple of trading days We broke out of that 180 S&A resistance the 50 S&A is honestly looking like it's about to cross the 180 S&A In a couple of days here if this uptrend continues meaning that you know whenever we get a 50 S&A cross Above the 180 S&A we've talked about this in a bunch of videos. That is a very bullish signal Very basic technical analysis here guys, you know, you know using simple moving averages to identify an uptrend Or a downtrend very basic guys, you know the 50 crosses above That's a bullish sign and of course if it crosses below like we saw back here in october That is a bearish sign So keep an eye on that potential cross that we might get in a couple of days And of course we're trading above that 180 simple moving average right now Like we've covered in the past couple of videos But if we take a look even closer here guys the continuation of the uptrend is still Evidence right it held above that red trend line that we can see here It's pushed to another higher high to around 27 24 and there's honestly no downtrend reversal pattern in sight I thought we were potentially getting it here a couple of days back with the double top pattern But we found the support and pushed to another higher high So that was invalid and of course I thought it was going to happen here again Potentially in yesterday's video that I talked about it. But you know, we saw the push to a higher high today So honestly guys the continuation is still there. It's not really looking like it's pulling back anytime soon Based off of the technical analysis. So now that we can see, you know the spx Let's take a look over here to the dow Jones to see what happened So we saw it was up around 175 points today Up around 0.7% just like the spx guys were trading above the 180 simple moving average with the 50 sma nearing the cross Above the 180 sma which does signal a lot a lot of bullish potential in that stock index etf Whatever we're looking at but in this case, of course, we're looking at the index of the dow Jones, right? And if we take a look on the smaller charts just like the spx guys, you know, we're still up trending We held above that uh trend line here We pushed to a higher high to about 25,200 dollars and there's no sign of a reversal in sight Right. I thought it might have been here with this potential double top. We didn't get it We pushed up we ended up gapping up pushing to another higher high I thought it could have been here potentially right this morning. I thought we were selling off But again, we held that trend line. We pushed to another higher high and uh, you know, the run is continuing So just to give you guys a quick little brief, um, you know showing of what You know the value of percent and what percentage value have we increased since this bottoming out point The dow Jones is up around 13.5% So that's a very strong start to 2019 and the spx I think it's up a little bit more probably around 15 right right around 14 or actually no right around 13 90 So close to 14% which again very solid start to 2019 and let's take a look at the nasdaq guys Of course the nasdaq is very tech heavy. That's why it's one of the higher moving indices today It's up around 1% and I thought at the close it might have been up around 1.1% But again for those of you guys that don't know these are the futures So they could be down a little bit after market hours in the reaction of google and let's see Yup, it did peak up here. We can see after google came out. They pushed down a bit So that's probably what is dragging down the nasdaq But uh, nonetheless, let's take a look at what it's doing on the 180 chart here We can see, you know, it got out of that horizontal pattern that we were talking about it broke the 6800 Resistance it held this one This previous resistance, which is now a new support as a support, right? It held it for a couple of days here We can see and now we're popping up above it So just like I talked about in the previous video guys, if you guys remember this comment down below But I was talking about how we were trading in this channel, right? We broke out of that channel Holding this new resistance old resistance as a new support with the possibility of filling this as the next channel, right? That's exactly why I have these trend lines here. It's pretty much playing out, you know, exactly as to how My technical analysis was done and what I said in the video a couple of days ago, right? So we can see the clear bounce on this support We broke out of that 6900 area, right? And now we're trending up potentially getting to around the $7,000 mark next and of course after that at around 7100 that is where we're going to see if the nasdaq futures Nasdaq indice in general is going to be able to, you know, fill that gap So as of now guys, you know, all of the indices are up trending still no sign of reversal and honestly the way it's looking guys With the optimism that's pumped into these stocks right now We could see some more green over the next couple of days, but i'm still sticking to my thoughts that We're going to pull back eventually and once we do pull back It's going to be pretty strong, right? Not saying we're going to have a recession There anytime soon, but you know, we have seen some hedge fund managers, including Ray Dalio say that they do see Signs of a recession potentially happening at the end of 2019 towards the beginning of 2020 But honestly, you could call bs on that because I feel like every single year over the past six years We've seen people say every single year that there's going to be a recession and there wasn't a recession But at some point, you know, it's going to happen guys and the later the later we wait the farther we get into this bull run You know the closer we do get to that recession, right? So now that we've got over, you know, the pretty much the the market update the brief analyses of the major indices Let's take a look at google stock very quickly. We'll talk about their earnings I have some notes here on my phone and we'll go over that briefly see the reaction of the price To the earnings report and then we'll talk about what I traded today and get on with the rest of the video. So Very briefly guys very very briefly here earnings per share beat by a heavy margin $12 and 77 cents in eps This quarter for google versus $10 and 86 cents Expected I believe the the drastic change in this is due to some tax reform plan I was reading about that I forget off the top of my head right now But for those of you guys that know what i'm talking about drop a comment down below But i'm pretty sure it's some tax law that made their eps stand out This particular quarter, but anyway the revenue was 31.84 billion versus 31.3 billion dollars expected Revenue rose 22 percent year over year their revenue growth benefited from strength and mobile search Including youtube guys. So one thing about google's earnings.  Well, obviously some of it's tied up in investments But the some of the lump sum of money that's in strictly cash, right? You know, they can go out and buy a different company. They can buy this buy that they have so much leverage Just having that much cash and I don't know how easily They can liquidate their investments whatever their investments are in but i'm sure you know They can drag some of that money out and put it into uh, you know, some other things if they decide to buy right? So one bad thing about uh, the google's earners report was that their operating income fell short of estimate 8.6 billion expected versus 8.2 billion dollars Reported so that's just a brief brief Gist of uh google's earnings report. There's a ton of things that I didn't touch upon But I just wanted to mention, you know the revenue growth The eps growth and of course the strength of the company with the 109 billion dollars in cash and investments And we can see guys google stock super super volatile here after market hours Spike from 1130 up to around 1160 That's a 3% move to the upside and then it swung down 4% nearly to the downside and now it's hovering around We see these two spikes here that does seem like a little bit of a manipulation there in google stock But uh, anyway, you know, it's down around three What's down like 3% right now after market hours not too crazy of a move to the upside or the downside in terms of google stock So let's talk about what I traded today very quickly before we talk about a couple other stocks etfs that do Look pretty solid right now and didn't move according to what I talked about in yesterday's video So what I traded today guys, I'm sure you all saw cron had an absolutely ridiculous run again today It was up 17 percent at one point. I captured around 1.2 percent of this Really early on in the day today guys I was done trading literally within 20 minutes of the market open and pretty much how this trade ended up going down Is very basic. We can see here We gapped up to around 21 dollars and 84 cents I was actually explaining this to someone in the dm on discord today. We gapped up 21 dollars 84 cents and ended up gapping down to around 21 40 where we found that support and that opened up Roughly 2% margin of profit and then once I saw guys what I wanted to see is a hold above that 180 sma As a new support we got that we saw a strong green candlestick It started to trend up and this is when I started to build in a position once we started to find that support Once we started to bounce back up That is where I ended up building my position right very very simple here I was looking to see if it was going to fill that gap from the pre-market high This is often how I day trade right, you know early on in the day within the first 30 minutes because A lot of stocks are choppy early on in the day This is an example of a big sell-off But I wanted to see if we found support it found support and I ended up getting in at around 21 dollars and 61 cents I believe and then I started to scale in until I took around a 1.2 Profit guys very quick little scalp trade up to around 21 90 the whole idea and goal of this trade was to profit on the gap fill and once we filled that gap guys I got my 1.2 Ended up taking my profits and then just pretty much played it safe for the rest of the day Sure, if I held you know, I would have made a crap ton of money I probably would have made 15 times the amount of money that I did make but You know profits profit consistencies consistency, and that's what I'm all about guys. I don't go for the home run trades I like to keep it very You know risk managed I like to keep it very consistent And I like to keep the wins and the profits small so that I can last You know over time because again like I say a lot in these videos If you go for the home run trade every single time and when I say home run trade These could be some of these penny stocks that are pumped up by some of these, you know stock people You know on the youtube and the internet that sell crap like that, you know the chat rooms It could be some other volatile penny stock that's on a bio farm, you know penny stocks some of these stocks guys They can go up 100 500 a thousand percent in a day But you can lose money if you get in late and you get in on the dump Let's say after you know the chat room whatever with these chat rooms I don't even know. I've never honestly been in one, but I heard a lot of bad stuff about them You know people losing money. Just be careful guys. Honestly, I'm here to you know Really help you guys out. I feel like you know a lot of people need to hear You know, it's not all about the home run trades You know you can lose money if you do get in at the wrong time on these penny stocks And that's just really discouraging You know a lot of people end up quitting due to that and honestly, I just hate seeing that it just gives you know traders And uh, you know stock trading in general a bad rap a bad name and you know, I'm not about that honestly I really hated that so um Let me know down below what you guys think about that But you know, that's pretty much it in terms of my trade today on kron Some other ones that i'm looking at and i'm really liking right now are you know these crude oil ETFs We finally got that pullback like I was talking about in yesterday's video that I was looking for We saw crude oil was a bit overbought I wanted to see that pullback and the hold above the e-m or the 50 sma rather We got that so that could potentially be or rather this could be potentially a solid entry point for uwt So uwt, I know a lot of you guys traded it today. I'm sure a bunch of you guys are swing trading it overnight This is one that i'm going to be looking to trade tomorrow possibly pre-market during the market We'll see how it ends up going so uwt guys. I'm watching that one very closely JD st ended up doing very well today. We saw it was up around three percent This is another one that I was talking about in yesterday's video I was looking for that pullback in the gold futures. We ended up seeing that pullback today from 13 30 Actually, no, is that around 13 25 earlier today? I believe right? Or no, that was at the close on this previous trading week and we saw it sell off all the way to 13 13 And now it's holding that support guys So let's say, you know, it breaks the support to the downside This could have continued to sell off maybe to the 1305 level But let's say it holds this as a support, you know, this could be a potential entry in jnug And this is exactly what I was talking about in yesterday's video guys and the beauty of these ETFs You can profit on the downside, which is what I pretty much called out, right? And then you can profit on the upside if you do find the support in the future that you are trading So that is what i'm looking at guys, you know tomorrow potentially entry point for JD st If we break here, but let's say we hold this support and we break this 13 20 resistance That could be another position to go along on jnug so Those are the two main ones that i'm watching tomorrow. We saw obviously the marijuana stocks I don't have to show you guys again. You saw cron did absolutely amazing today. We saw cgc It popped up to like 51 dollars at one point. Um, we saw abc This one did very well not abc. This isn't it. Wait, is this it? No, no, no, this is not an acb ACb is that a roar? Yeah, there it is acb. That's up 9% today guys So, you know, these are all screaming pullback, but every time I do say that they run up another 10% So honestly, they're disobeying the technicals guys.  So i'm waiting for the crash. It's going to come You know, we saw it this time right a crash from 12 all the way to like 10 50 in the matter of about a day It seems like so we could potentially see that and one of these days coming up. Who knows so Another one I want to talk about briefly before I end the video is j and j This one also got the pullback that I was waiting for and it's looking like it's holding above that previous resistance Which is now a new support at around 131. Sure, it could probably sell off more to around 131 50 And if it does that might be an even juicier entry point for a swing trade But of course if we break below, you know, that's going to be a break of pattern We could be heading back to the downside So i'm watching to see if it does end up holding this pattern guys This could be a good entry point for a safer swing trade in my opinion So I hope you guys enjoyed the video if you did feel free to drop a like leave a comment and subscribe And i'm sorry if this fan is bothering you my computer is overheating right now and the fan does seem a bit loud So if you got if you got that in the audio and it bothered you I do apologize You know i'm in the process of getting a mac book pro right now, which will not have these overheating issues But until then guys, i'm sorry if you do hear these, uh, you know this fan in the videos And I hope you guys did enjoy it like I said drop a comment leave a like subscribe again Thanks for watching guys. I really do appreciate you all. I'll catch you later


Hello everyone, this is Alice Gao. In this video, I will describe the lowest cost-first search algorithm. Technically, LCFS is an uninformed search algorithm because it doesn't make use of the heuristic function. So far, the uninformed search algorithms focus on finding any solution. There's little guarantee on the quality of the solution found. DFS has no guarantee. BFS and IDS are guaranteed to find the solution with the fewest edges, but this path is not necessarily the one with the least total cost. If all the edges have the same costs, the solution is optimal. What if our goal is to find the optimal solution? One way to do this is to keep track of the costs of the path found so far. This idea takes us to the lowest cost-first search algorithm. LCFS maintains a frontier, which is a priority queue ordered by the path costs. It always selects the path with the lowest total cost to remove from the frontier. You may have learned the Dijkstra's shortest path algorithm, which is similar to LCFS. Let's trace lowest cost-first search on a search graph. We'll break ties as follows. If multiple paths have the same cost, we will order the path by the last node and choose the one that comes first in alphabetical order. Pause the video and trace the algorithm yourself. Then keep watching for the answer. This is the search tree when the algorithm terminates. I will describe the tracing process in a separate video. Let's look at the properties of LCFS. First, the space and time complexities for LCFS are both exponential. LCFS generates all the path whose cost is less than the cost of the optimal solution. The number of such paths could be very large. It is not doing well with space and time because we're doing some intensive optimization to make sure that we find the optimal solution first. As we start talking about more complex search algorithms using the cost and the heuristic functions, it becomes more challenging to characterize the time and space complexities. It is also less interesting to discuss these complexities because they tend to be exponential in the worst case. Let's look at completeness and optimality.  The first condition is that the branching factor must be finite. The algorithm may decide to expand all the children of one node before going deeper into the tree. Having a finite branching factor makes it possible for the algorithm to explore all the children of a node if necessary. The second condition is that the cost of every edge is bounded below by some positive constant. In other words, the cost of an edge cannot be arbitrarily small. Without such bounds, there could be infinite path with a finite cost. The Xenos paradox devised by Aristotle illustrated the idea of this case. Suppose that we have edges with costs 1 over 2, 1 over 4, 1 over 8, 1 over 16, and etc. We can construct an infinite many number of paths using these edges and the cost of every path is less than 1. The problem here is that we would be exploring these paths forever and never reach a path of cost of 1. In general, LCFS is cautious and conservative. It looks at all of the potential paths so that it doesn't miss the optimal solution. That's everything on the lowest cost first search algorithm. Let me summarize. After watching this video, you should be able to do the following. Explain LCFS at a high level. Trace LCFS on search graph. Explain the properties of LCFS. Space complexity, time complexity, completeness, and optimality. Thank you very much for watching. I will see you in the next video. Bye for now.


Hi everyone now today. We're doing a lip implant. So we look at the implant. It comes from permafacial lip implant It's a silicone implant, but the thing is just like a breast implant. It's in a covering You could see how it has a shell, but it's very flexible. We're going to put it in baited dine We're going to clean it antibiotic wise just like that So you could see it's all in a gel form So it's not loose like the type that we in you would inject you see it's all in a piece You can bend it. You can laugh. You could cry whatever It stays in there. So what we're going to do we've already done the dissection our patient is numb What we're going to do is we're going to feed it to ourself and we do the shoe polish thing To make sure it's centered. You see how i'm doing this called the shoe polish maneuver Boom, so we know it's going into centrally And then we let go and then we pull and boom It's in there So you could see the dissection is done already our patient is awake any pain? No, no pain. No pain She's comfortable the dissection is done She was nervous because she had seen some videos, but I told her after I numb her up. She won't feel anything She's been doing great. We've been chit chatting. She's doing good We're opening up going to do the lower part and then the buckle fat. It's going to be a fun day here So here's again our perma perma facial implant 5 millimeters 65 millimeter 5 millimeter is the thickest they have we checked the expiration date. It's ready to go You know it's from texas. Here it is.  Look at that beautiful shoe shine maneuver. You see it use the ends are equal Right there you maneuver it back and forth. How you doing? Good Okay, and then we just pull it in and bloom it springs into position So here it is. Here's her buckle fat in record time. I think it took me a home 30 seconds to find it in there and boom Not even 30 but look at that beautiful buckle golden fat right there We take out all of it as much as possible. That's what she said. She told me make sure dr. Saul you take all of it out. What do I do?  There's the one from the right look for a nice girl like this. That's a big one That's a really juicy one. She's gonna see a nice difference I don't know if you guys see it on my glove or not, but here it is. There it is. Look at that Here's the one from the left side. You can see the buckle fat. It's all removed.


So for us annotation does many things but mostly it really breaks down that language barrier because when you come across a word that's not intuitive. And you really, you know a lot of scientific words, you can't even really use context clues to find out, you know, what they mean. So we've done a ton of work with the annotations and just defining vocabulary words as a way to just encourage people to keep reading, you know this word might trip you up and you're going to stop but here's that definition. So keep going.


Hi, this is George Cowell and today in this short video, I want to talk with you about when you are making an offering of your services, when you're trying to announce your services, one of the mistakes that people make is that they don't announce it more than once or maybe twice. They think, gosh, I didn't get much response, so therefore, nobody wants it. Now, in some cases that may be true and you need to modify your services, but before you have repeated it enough, simply the idea is you might not have the people, so typical, okay, so there's been a well-known that people need to see something multiple times before they make a decision. Now, I've seen studies showing that people need to see an advertisement 13 times before they make a decision, and recently I saw it, it was seven times, but regardless, the point is that people need to see it more times than you're probably announcing it, okay, and that's the important point. And when you are announcing it, for example, on social media or to a group of people, like to your email list, announce it at least three times in a short time frame, like within one month, announce it at least three times so that they have a chance to so that your offer has a chance to make an impression on them so they can make a decision. When you're announcing it to one person, you're trying to reach out to one person, now, yes, you could reach out multiple times, but it could get annoying for them. So let me give you one more principle, which is to be in service when you do the repetition, to be in service. So whether it's announcing it publicly or whether it's announcing it one person, always be asking, how can this thing be of true service to the people I'm announcing, and how can I announce it in a way that's of true service? So for example, you might emphasize that you're giving something away for free as part of your launch of your service, okay, maybe you're giving a free call or some other benefit, some other gift, or you approach it from, hey, if you know someone who is really struggling with this issue, I want you to know that I'm now offering the service, and it really will be helpful for them. So feel free to have them get in touch with me, and I will help them as much as I can. So be in the place of service when you're announcing it and when you're repeating it. And the last thing I want to say is longevity. There is repetition in the short term, which is within a month, you announce something three to five to seven times, okay? And again, within service and try to make it a variety of things, I'll put more in the notes of this video to give you some ideas of variety, the ways to announce it, but yeah, in service and with some variety. But also there's longevity. So people may see your launch and they may see the three to five announcements about your thing. Maybe they only out of seven times you announce it, maybe they only saw three times, okay? But they just are getting, okay, they're starting to think about your services, and it takes people time. Sometimes it takes people months or years before, and this happens to me all the time, someone's been following me for years and they find, okay, now it's finally time. People have different lengths of the sales cycle, you might say. So longevity is important. So keep announcing it. You do a launch and then you do another launch a couple of months later, and then you do another launch three to six months later, okay? The longevity will allow the people who need more time to then make a decision to work with you or to refer you. So I hope that's helpful. As always, I'll put more notes underneath the video or in some cases above the video. And until the next video, I wish you confidence and clarity and from a place of service, repeating your services enough so people can really make a decision. Take care.


I'm Salvatore Bobona, and today's lecture is Major Schools of Political Economy, in theory and in practice. This is an introductory-level lecture meant to introduce students into basic concepts of left and right in political economic discourse. The field of political economics spans all of the social sciences and public policy as well. We might define political economics as the study of production, distribution, and exchange at the level of an entire economic system, whether that's a country or even the entire world as a world system. Unfortunately, the term political economics has the word economics in its name. Political economy is not about economics. It is about the economy, but it's about how economic rewards are distributed or allocated in political ways. It's not about how to economize, and it's not about the discipline of economics. In fact, political economy and political economics are studied by anthropologists, geographers, political scientists, and sociologists, and yes, even some economists as well. Questions of production, distribution, and exchange are always political. The question of who gets what can never be divorced from a political decision based on the political rules of the game. There's no such thing as a neutral or natural economic space. All economies are socially and politically constructed. In the political economy or any political economy, different players in the system attempt to enact rules or tilt rules in their favor. We usually think that certain ways of thinking about the game favor different kinds of players in the game. We usually think of a kind of playing field that tilts either left or right. Broadly speaking, the left is associated with pro-union, pro-uneducated, pro-worker, pro-poor kind of approaches, pro-people, if by people you mean the unwashed masses, versus the right wing is associated with pro-business, pro-educated, pro-owner, pro-rich, pro-elite points of view. All the kinds of policies that are generally pursued by the left include minimum wage policies, collective bargaining, lots of government regulation of the market, progressive systems of taxation, and that can be contrasted with the kinds of policies that are usually associated with the right. Getting rid of minimum wages, having individual bargaining, small government, a flat or even regressive tax systems in which poor people pay more than rich people. At the theoretical level, there are different theoretical systems that tend to build a case either for the left or for the right or for somewhere in between. There are many theories of political economy, but we can perhaps identify five big groupings, one around a bunch of posts, post-development, post-growth, eco-feminist, post-modern approaches on the very far left. Then classical Marxists, left-wing social democrats, kind of as the center left. Institutional economists in the center, mainstream neoclassical economics as a profession, business-friendly economics in the center right, and then at the far right, market fundamentalists who want all the rewards to go to whoever is able to garner the most power in the market. Taking each of these in turn, first, post-modernists may be questioned the relevance of the entire field of political economics. They're so far to the left that they're off the playing field entirely. Political economics, they might believe, focuses attention away from what's really important. We shouldn't even be fighting over money. We should be getting rid of money entirely. The key insight of people on the far left, these post-modern approaches, is that the very field, the very discipline of political economics reinforces a patriarchal social system that focuses on the concerns of usually men, certainly of men as a gender class, and especially of powerful men. The idea here is that once we start talking about who gets what in a political system, the battle is already lost. There shouldn't even be a system for deciding who gets what. Of course, a key criticism of this approach is that if you try to exempt yourself from the system, if you don't fight the battle, then you'll simply be slaughtered by those who will fight the battle. Key names in post-modern approaches, Wolfgang Sachs in Sociology, Susan George in Environmentalism, Claudia von Fehrloff in Eco-Feminism, these are all people who would be on the post-modern far left of the political economic spectrum. Marxists and neo-Marxists, or what might be called social democrats, claim that economic systems serve specific class interests. They would say that there are many possible economic systems. Capitalism is only one of them. We could have a better system. We could have rules of the game that benefit ordinary people more than they benefit the rich and the powerful. The key insight on the Marxist left or in the center left is that market outcomes are the result of non-economic struggles. Power in society percolates through the market system to reward some groups of people over others in the market. A key criticism of Marxian approaches to political economy is that it's hard to see how classes actually operate together. If it's really the proletariat versus the bourgeoisie, where are the bourgeoisie having their meeting in which they decide how they're going to oppress the proletariat? Key names associated with Marxian approaches are of course Karl Marx himself, but also contemporary figures like Immanuel Wallerstein and the geographer David Harvey. In the center we might find institutional economists, the most recently, most famously in recent memory, Barbara Ostrom, who is a political scientist who won the Nobel Prize for Economics for her centrist institutional economics. Institutional economics as a discipline emphasizes that economic activities don't occur in a vacuum. The government has a role in regulating society, and not just the government, but civil society does as well. The key insight is that institutions matter. Key criticism is that institutions are only defined as good if we like the outcome. I mean it's hard to disprove an institutionalist's claim, it's hard to say in advance what institutions will turn out to be good ones. Key names in institutional economics, I mentioned Barbara Ostrom, she had not yet won the Nobel Prize when I created these lectures, but also Manker Olson, Douglas North, Darren Asimoglu. Neoclassical economists usually use formal theoretical models based on a rational choice understanding of how people behave. Neoclassical economics is the center right, the respectable right wing of the political spectrum. Neoclassical economists believe that prices are ultimately driven by supply and demand. If you've taken economics 101 in a conventional economics department, you've studied neoclassical economics. Now a key insight is that people do in fact respond to economic incentives. It is true that people are rational utility maximizers. The key criticism is that that's not all they are. People have much more to them than just economic rationality, and in fact economic rationality may only be a small part of what people are about. Key names in the development of neoclassical economics are people like Vilfredo Pareto who developed the concept of Pareto optimality on which all of neoclassical economics is based, the economist Alfred Marshall, contemporary economists like Robert Solow. On the far right of the spectrum are market fundamentalists, sometimes called neoliberals, though neoliberalism means many different things as well, but market fundamentalists who argue that markets aggregate knowledge and markets know best. We should always trust the decisions and the allocations that are made in markets because markets know more than any of us do as individuals. The key insight is that market prices do contain information. If you want to know what is scarce in the world, look to market prices. If market prices are rising, then something has to be economized on. If a market price is falling, it means that the world has too much of that thing. But a key criticism is that markets are prone to booms and busts, to bubbles. There's lots of follow the leader behavior. Market prices don't always accurately reflect the world. If market prices accurately reflected the world, then we would see the world advancing every time the stock market went up 20% and we would see the world coming to an end every time stock markets collapsed. Key names associated with market fundamentalism, Friedrich Hayek, Milton Friedman, and Alan Greenspan, the chair of the United States Federal Reserve Board at the time of the global financial crisis in 2008, sorry, just before the global financial crisis in 2008. These different philosophies, these different academic and intellectual underpinnings lead to different policies being implemented in practice. And here I'm going to turn to the geographer Patrick Bond. Now Patrick Bond is himself a Marxist or maybe even a post-Marxist philosopher on the very far right, far left end of the spectrum, but his categorization of these five approaches and how they translate into policy has become, I think, a standard way of looking at the world. Patrick Bond identifies five parallel ideologies in contemporary political economic practice. And he divides them into two types, the center and left, which are civil society perspectives that place an emphasis on people, and the right and far right, which instead put an emphasis on control, on governing, on governmentality rather than organic governance by the people for the people. He calls these five approaches the global justice movements, third world nationalism, post-Washington consensus in the middle, Washington consensus, and a resurgent right wing on the right. Global justice movements correspond to those post-approaches to political economy, post-modern, post-feminist approaches to political economy that operate outside the sphere of organized national politics. This is a photo I took at Occupy, Wall Street, Washington, D.C. encampment. Global justice movements don't usually try to seize political power to promote a program. They're arguing that we need a whole new configuration of politics. On the center left, third world nationalism has been a movement to resist the domination of rich countries, and this is usually inspired by Marxian political thought. Take of Chavez's Bolivarian Republic in Venezuela, the Marxian movements in Ecuador and Bolivia, even the Workers' Party in Brazil, which under Lula da Silva, before it collapsed in a corruption scandal, under Lula da Silva had promoted policies that dramatically improved the well-being of poor people in Brazil. Third world nationalist movements usually have a Marxian approach to resisting global economic forces for the benefit of ordinary people. In places like in France and in Germany, this takes a form of not so much a third world nationalism as a social democracy, a popular social democracy. As I am speaking in early June 2016, in France there are major strikes nationwide demanding better working conditions for ordinary people. The post-Washington consensus is based on the dominant centrist view of the economy as something that is shaped by institutions. Another manifestation of that has been flexibility policies. People arguing that the best kind of labor market is one underpinned by a generous welfare system that allows people to work as independent contractors because they know they can always fall back on the welfare system when necessary. All sorts of such arrangements for getting the institutions right, whether it's flexibility in the Netherlands and Denmark or other policies in other parts of the world, the nudge policies that have been popular in the United States and United Kingdom, these are all centrist institutionalist approaches to making day-to-day policy that broadly benefits people by pushing their individual behaviors in directions preferred by the people coming up with the policies. On the center right is the Washington consensus. This was perhaps the dominant policy approach in the 1990s whereas the post-Washington consensus is maybe then the dominant approach of the 2000s. The Washington consensus of the 1990s was that markets should be open. There should be liberalization that markets, if not no best, at least markets are the best way to serve people. There should be less government regulation. There should be free, open currency exchanges and lower taxes on the wealthy. This center-right set of policies that spread around the world. Here's an example of the Washington consensus in action when countries like Latvia and the other countries of Eastern Europe became independent after 1989. They generally implemented the kinds of policies promoted by the Washington institutions like the International Monetary Fund. As you can see, they ended up with tax systems that taxed low wages much more than they taxed high wages. So we can compare the wage taxation in Latvia, which very quickly goes up from 0 to 46 percent with the wage taxation system in Australia, which starts out very low.  The Latvian policy represents much more the Washington consensus, whereas Australia maybe represents more the post-Washington consensus. And finally, the resurgent right wing has promoted markets above everything else. A key example might be the California natural gas market liberalization. In California in the early 2000s, natural gas was simply made a tradable commodity, resulting in enormous price swings as private companies most prominently Enron, which later went bankrupt and not only went bankrupt but the heads of it were prosecuted for fraud. The idea was that private companies knew best how to deliver services and markets like the California natural gas market were simply cornered by corporations who were free to do whatever they wanted in a completely open market but an open market in which they had enormous power. So five perspectives, five sets of policies. I said before the 1990s, maybe in the 1970s, those Marxian inspired third world nationalist social democratic center left was dominant. By the 1990s, the Washington consensus, the center right had become dominant. In the 2000s, we even saw a brief swing towards market fundamentalism and the far right. A lot of people think we've made me move back towards the center right, maybe even towards a centrist post-Washington consensus. But I think that these left right, this left right spectrum of political economics theories and political economics practices is actually becoming obsolete as the 21st century develops. Instead I've started talking about the emergence of a new theoretical and policy era, what I'm calling the American Tiansha. Tiansha is a word I'm borrowing from Chinese international relations theory. It means all under heaven or essentially a global world system. This American Tiansha, this American world system is no longer one of left and right because it's instead a system of center and periphery. The American Tiansha is a cultural system that values adherence to Anglo-American liberal norms above all else. So the important thing is not whether someone is left or right, whether someone is a poor worker or a rich industrialist. The important thing is to be respected and to have a place in the Anglo-American cultural system. Thus American workers who are organized in unions in the United States are powerful in a way that maybe capitalists in developing countries are not powerful because those American workers have access to the levers of the American state machinery. The political economy of the system thus becomes a scramble for power in the American system as opposed to a battle of left versus right. It's a battle of center versus peripheries or maybe center versus periphery single, one big outer edge of the world. This new configuration is beyond left and right. We've seen a shift because we no longer live in an economic system. Left and right are terms that refer to the world market. The whole premise of left and right political economics is that there's a world market with a playing field that can be tilted either towards left or towards the right. I think what we see now instead is not a world market but a world polity, a political system that is apportioning rewards from the center and as a result instead of left, right we see a shift to center periphery. I would say that we no longer live in a political economy. We now live in an economic polity and what we really will need in the 21st century is a new science of economic politics. Thank you for listening. You can find out more about me at my website, salvaturbabonus.com, where you can also sign up for my monthly newsletter on global affairs.


little excursion in the grass and there's something hidden in there that was quite hard and we've hit it and it's popped one of the axles out of the out of the diff so we're going to make some quick repairs here and we'll solder on. We'll miss maybe two stages but we'll link in somewhere along the line and persevere. It's an endurance event. The plan this morning was to chase a couple of guys in front because they weren't too far away but that's out the door now so the plan now is to finish. We've been plagued by issues the whole rally so this is just just another one of those things that's rallying.


Welcome to lecture 17 on measure and integration. Today, we will start the topic of integration. First, I will explain the building blocks for the integration and how the process will be done. So, the topic for today's discussion is integral of non-negative simple measurable functions. See, the basic idea is we want to define the notion of integral for a function f defined on a set x taking values in r star. So now, for a function f, we can represent a function f as the positive part minus the negative part. The advantage of doing this is that f plus and f minus both are non-negative functions. So, it is an integration being a linear process. So, integral of f is going to be equal to integral of f plus minus integral f minus. So, it is enough to define the notion of integral for non-negative functions. And for non-negative functions f on x to r star, we recall that we can take it as we look at functions which are first of all very simple functions. For example, let us look at a function f which is an indicator function of a set x. Now, of a set a, a contained in x. This is a function which takes only two values. So, indicator function of a is a function on x taking values in r star. So, chi of a at a point x is equal to 0 if x does not belong to a and is 1 if x belongs to a. So, one can think of this function taking only two values. Now, the value where it is 0, the integral. So, we want to define the notion of integral. And this is going to be with respect to a measure mu on x. So, mu a measure on subsets of x. So, we are going to write it as x a d mu. So, what it should be? On a the value is 1. So, we will like to put it as 1 times mu of a. In some sense, mu of a is the size of the set and 1 is the height. So, this is in a sense the area of the graph of the function. So, let us look at functions which are going to be linear combinations of indicator functions. So, we start looking at the integral of non-negative simple measurable functions. So, let us recall. So, we will fix our notation that from now onwards, we are going to work on a measure space x s mu, where x is a set, s is a sigma algebra of subsets of x and mu is a measure on defined on s. And this is a complete measure space. That means that all sets a says that mu of a is 0 implies that a and all its subsets are inside s. So, let us denote by l lower 0 upper plus to be the class of all non-negative simple s measurable functions on x. So, now let us recall what was a non-negative simple measurable function s. It is a function defined on x taking non-negative values and it has a representation s of x is equal to sigma i equal to 1 to n, a i times the indicator function of the set a i evaluated at x, x belonging to x, where a i is a 1, a 2, a 3, a n are extended real numbers and the sets a i's are in the sigma algebra s. So, they are in the sigma algebra a i's are in the sigma algebra s and they are pair wise disjoint. That means a i intersection a j is empty for i not equal to j and the union of these sets is equal to x. So, this is going to be the class of non-negative simple measurable functions. For such class of for functions in this class, we are going to define the notion of integral. So, for a function s in this class, if its representation is as given before. So, if s of x is equal to sigma a i indicator function of capital A i then its integral is defined as s d mu. So, integral is noted by integral sign s of x d mu x to be a i that is a value of the function on the set a i times the measure of the set a i mu of a i. So, integral of s with respect to mu as written here is defined as sigma a i times mu of a i. a i is the value taken on the set a i. So, a i times the size of the set a i. So, mu of a i. Sometimes we do not indicate the variable x, we just write as integral s d mu to be the integral of the simple function s non-negative simple measurable function s with respect to mu. And let us note here that our representation the integral is with respect to a representation of the function. So, first of all we would like to show that integral s d mu is well defined. So, let us prove that the integral is well defined. So, let us take a function s belonging to L plus 0. So, it is a non-negative simple measurable function. So, let us say s is written as sigma a i indicator function of a i i equal to 1 to n also representable as sigma j equal to 1 to m of some b j chi of b j, where the sets a i belong to the sigma algebra s, b j belong to the sigma algebra s and union of a i is equal to x and union of b j is also equal to x and these sets are disjoint. So, a i intersection a j empty and b i intersection b j is empty for i not equal to j. So, let us say the set s a simple function s has got two representations possible. So, what we want to show? So, we want to show that the integral of s. So, integral s d mu is well defined and that means what? So, mathematically that means we have to show that sigma a i mu of a i 1 to n is equal to sigma j equal to 1 to m b j mu of b j. So, this is what we have to show. So, let us start. So, sigma a i mu of a i i equal to 1 to n I can write it as sigma i equal to 1 to n a i and then mu of this a i can be written as union of a i intersection b j j equal to 1 to m because union of b j is equal to x. So, a i intersection x and that is same as this. Now, this is a b j are disjoint. So, these sets are a i intersection b j for i fix are disjoint. So, by using finite additive property of the measure we have this is equal to i equal to 1 to n a i and this is nothing but sigma j equal to 1 to m mu of a i intersection b j. Similarly, we can write the other side that is sigma j equal to 1 to m of b j mu of b j to be equal to sigma j equal to 1 to m b j sigma i equal to 1 to n mu of a i intersection b j. So, the left hand side here is written as this sum, the right hand side is written as this sum. Now, we want to show that these two sums are equal. Now, let us observe that given that the function s has got two representations this equal to this. So, how is this function calculated? At a point x if x belongs to a i the value is a i and on the other hand it may belong to some b j the value will be b j. So, that forces one to say that if x belongs to a i intersection b j then a i must be equal to b j. So, this is the crucial thing to note here that if s a non-negative simple measurable function is given two representations one is sigma a i capital a i indicator function of a i and sigma b j indicator function of b j. Then for x belonging to a i intersection b j the value of s of x on one hand it is a i other hand it is b j. So, a i must be equal to b j. So, this is the crucial thing to note. So, let us make this observation and write it out. So, note that if x belongs to a i intersection b j if x belongs to a i intersection b j then s of x is equal to a i and is also equal to b j. So, a i is equal to b j and if x does not belong to a i intersection b j then s of x is equal to 0. So, that means in this summation whenever x belongs to a i intersection b j this a i is going to be equal to b j. Otherwise in this sum that term does not matter. So, that proves the fact that so that will imply from these two equations from equation 1 and equation 2. So, this implies from equation 1 and 2 that sigma a i i equal to 1 to n of mu a i is equal to sigma j equal to 1 to m b j mu of b j. So, that is integral s d mu can be defined as either of these sums. So, is equal to either this or this is well defined. So, the integral of a non-negative simple measurable function. So, we can choose any representation of we can choose any representation of the non-negative simple function and define its integral in terms of that. Next, let us look at properties of this integral. So, we are going to look at functions s s 1 s 2 which are non-negative simple measurable functions. Alpha will be a real number, alpha bigger than or equal to 0. Then we are going to look at what happens to various properties of. So, first observation is that integral s d mu is a non-negative number. It could be equal to plus infinity. So, integral s d mu is an extended non-negative real number.  So, this is a non-negative number. So, this is an obvious property. The second property we want to check that for a non-negative simple function s alpha s belongs to l plus 0 plus and the integral of alpha s d mu is same as alpha times the integral of s d mu. So, let us check that. So, s belongs to l plus 0 is a non-negative simple measurable function. So, let us write s is equal to sigma a i indicator function of a i, where union a i is equal to x. So, whenever it is a partition, we will write as this square bracket union over i equal to x and alpha is a non-negative alpha belonging to r star, alpha bigger than or equal to 0. Then, alpha of s has the representation. It is alpha a i chi of a i and a i is still a partition of x. But that means, if this is the representation, so integral alpha s d mu, integral of alpha s with respect to mu, is it going to be equal to by our definition i equal to 1 to n alpha a i times mu of a i. This is a finite sum, non-negative everything. So, alpha comes out, alpha times the summation of i equal to 1 to n of a i mu of a i and that is nothing but alpha times integral of s d mu. So, that proves the property that the integral of non-negative simple functions is, if you multiply it by a constant alpha, then the alpha comes out. So, integral of alpha s d mu is equal to alpha times s d mu. Next, we want to show that it is a linear operation. So, we want to check that if s 1 and s 2 belong to L 0 plus, then s 1 plus s 2 belong to L 0 plus. Actually, we have already checked, but we will check it again today also. The integral of s 1 plus s 2 d mu is integral of s 1 plus integral of s 2. So, for such things, let us take a function s 1, s 2 belonging to L plus 0. So, non-negative simple measurable function. So, let us write, let s 1 be equal to sigma a i chi of a i, where a i is form a partition of x. Let us write s 2 as sigma j equal to 1 to m b j chi of b j, union b j is equal to partition of x. So, if you recall, we had said that we can bring both s 1 and s 2 a common partition and what is that common partition? a i intersection b j. So, what we are saying is, we can write s 1 as sigma i equal to 1 to n, sigma j equal to 1 to m, a i chi of a i intersection b j. Similarly, s 2 can be written as i equal to 1 to n, sigma j equal to 1 to m of b j chi of a i intersection b j. Now, here note that union over i and j a i intersection b j, that is a partition of the whole space. So, that is equal to x.  Now, we can define what is s 1 plus s 2. So, s 1 plus s 2 is going to be equal to sigma over i 1 to n, sigma over j equal to 1 to n a i plus b j chi of a i intersection b j. That is clear because on a i intersection b j, s 1 is a i and on a i intersection b j, s 2 is b j. So, s 1 plus s 2 will be equal to a i plus b j on a i intersection b j. So, once we have got on a representation of s 1 plus s 2, we can define what is the integral of s 1 plus s 2. So, this representation gives us that integral of s 1 plus s 2 d mu is equal to summation over i 1 to n summation over j 1 to m of a i plus b j into mu of a i intersection b j. So, because this is the representation, a i plus b j is a value on the set a i intersection b j. So, the integral is going to be equal to summation over i summation over j of a i plus b j, the value on the set a i intersection b j. Now, the right hand side, we can write split. So, that is equal to two terms. One is summation over i summation over j of a i times mu of a i intersection b j plus the second term summation i equal to 1 to n summation j equal to 1 to m of b j. So, a i and second term is b j mu of a i intersection b j. And now, these are all finite sums. So, we can write the first term as sigma i equal to 1 to n. Take a i outside and this is summation of mu of a i intersection b j, because this is summation over i only. So, we can take it out over j equal to 1 to m of a i intersection b j plus here summation over j and summation over i. So, we will write it as summation over j first b j and inside is summation over i equal to i. I have interchanged the order of summation. They are finite terms only, finite sums only. So, that is allowed. So, that is 1 of mu of a i intersection b j. Now, we observe that the first sum by the finite additivity property of the measure is nothing but mu of a i and this summation over i, this sum is nothing but mu of b j, because a i is form a partition of x and here b j is form. So, first term is equal to summation i equal to 1 to n a i mu of a i plus summation j equal to 1 to m b j of mu of b j. Clearly, this is integral of S 1 d mu plus this second term is integral S 2 d mu. So, that proves the fact that integration is a linear process. If S 1 and S 2 are in L 0 plus, then S 1 plus S 2 also is in L 0 plus and the integral of S 1 plus S 2 is equal to integral of S 1 plus integral of S 2. Next property we want to check is the following that for a set, if E is a set in the sigma algebra S and we multiply S non-negative simple measurable function by the indicator function of E, then that function also belongs to L 0 plus that again we had checked it earlier when we defined non-negative simple measurable functions. So, its integral is defined and we want to check that E going to nu of E which is integral of S indicator function of E d mu is actually a measure on S. So, this gives a method of generating more measures on the sigma algebra E. So, let us prove this property. So, let us take a function. So, let us take a non-negative simple measurable function L plus 0 S of given by sigma i equal to 1 to n A i indicator function of A i where union of A i is equal to x and E is a fix set in the sigma algebra S. Then S times the indicator function of E. So, multiply this equation on both sides by indicator function that is i equal to 1 to n A i chi A i multiplied by chi of E and now here is the observation that the product of indicator function of two sets is nothing but the indicator function of the intersection. So, this can be written as i equal to 1 to n A i. This product indicator function of A i into indicator function of E can be written as the indicator function of A i intersection E. So, that is the only observation one has to make. Now, S times indicator function of E is given by this. So, where union of A i's intersection E what will be that? That is the disjointed union giving you the set E and on E complement this function is 0. So, if you like you can add one more term here 0 times the indicator function of E complement, but that is not. So, normally whenever that kind of a set that term will not mention it here. So, automatically on the complement it is 0 and that gives a partition of the set. So, this means S of indicator function of E is A i times indicator function of A i intersection E where these things form a partition. So, that implies S times the indicator function of E is a non-negative simple measurable function and what is the integral of that? So, integral of S chi of E d mu is equal to sigma i equal to 1 to n A i mu of A i intersection E. So, that is the integral of this function. So, we want to prove that if we call this as mu of E that is a measure. So, let us check that property to check it is a measure what we have to check. So, mu of mu of a set E is defined as sigma by our previous calculations A i times mu of A i intersection E i equal to 1 to n, where A i's are partition of X and A i's are in the sigma algebra always. So, claim mu is a measure. So, what is due to be checked? mu of empty set equal to E is empty set. So, mu of A i intersection E that is empty set. So, that is 0. So, it is equal to 0. What is the second property? We want to check mu is countably additive. So, for that let E be equal to union of E j, j equal to 1 to infinity, where all the sets are in the sigma algebra. So, you want to show that to show mu of E is equal to sigma mu of E j, j equal to 1 to infinity. So, that is what we have to show. So, let us compute both sides and show the required property. So, let us look at mu of E. So, mu of E is equal to sigma i equal to 1 to n A i mu of A i intersection E. By definition mu of E is defined as this thing. What is E? Let us put the value of E. So, it is i equal to 1 to n A i mu of A i intersection union, disjoint union E j, j equal to 1 to infinity. That is by the definition of by the fact that E is a disjoint union of E j. But, that we can write it as summation i equal to 1 to n A i mu of. So, this is nothing but, so we can write as disjoint union over j 1 to infinity of A i intersection E j by the distributive property of intersection over union. So, this is a countable disjoint union of sets in the sigma algebra. So, by the countable additive property of the measure mu, this term is equal to summation i equal to 1 to n A i summation j equal to 1 to infinity of mu A i intersection E j. Now, note that we have got two sums here. One is summation A i, another is summation j equal to 1 to infinity and all are non-negative extended real numbers. So, we can interchange the order of integration without any problem. So, we can write this as summation over j first, then summation over i 1 to n A i mu of A i intersection E j. So, we write this as this. Now, note that this the term summation over i A i mu of A i intersection E j is nothing but, the mu of E j. So, by definition this is summation over j equal to 1 to infinity. So, this is mu of E j. So, we have shown that mu of E is summation mu of E j is whenever E is equal to union of disjoint, prepare wise disjoint sets E j. So, that proves that mu is a measure. So, we have proved this property also that for a set E in S, the integral S times indicator function of E is a non-negative simple measurable function and if it is integral is denoted by mu of E, then mu of E is a measure as E varies over measurable sets. This measure has a very nice property. So, this mu measure mu of E has a very nice property that mu of E is 0 whenever mu of E is 0. So, let us just check that property again, check that property. The mu of E is defined as summation i equal to 1 to n A i mu of A i intersection E, where union of A i is equal to x. So, if mu of E is equal to 0, that will imply that mu of each A i intersection E is also 0, because A i intersection E is a subset of E and mu is a measure. So, mu is also monotone. So, mu being monotone, mu of A i intersection E is less than or equal to mu of E, which is equal to 0. So, that means this is equal to 0. So, implies mu of E. So, each term in the definition of mu of E is 0, that means mu of E is 0. So, this new measure, which is defined via integration of non-negative simple functions has the property that mu of E equal to 0 implies mu of E equal to 0. This is a very special property. So, it relates two measures mu and mu. That means it says whenever E is a set of measures 0 for mu, it is also a set of measures 0 for mu and later on almost in the end of the course, we will characterize such measures. Whenever two measures are related by this, there is a theorem, which says that mu must be representable as integral with respect to mu. So, we will come to that theorem a bit late in our course when we have finished integration and some more properties of it. So, this new of E, which is written as, which is an integral is having a special property. And let us also mention that integral of S indicator function of E d mu is also written as integral E of S d mu. So, this is another way of writing. So, this is called integral of S over E. So, we say this is integral of S over the set E. So, that is the notation will follow because outside E, S is 0 in this representation. So, next property we want to check is that if S 1 is bigger than S 2, then integral S 1 is bigger than integral S 2. So, let us check that property. So, let us write S 1, which is non-negative simple measurable function as sigma A i indicator function of A i and S 2 as sigma B j chi of B j, j equal to 1 to m. So, as we had mentioned, whenever you want to do some analysis regarding two simple functions S 1 and S 2, bring them to a common partition. So, we will write this is also equal to sigma over i 1 to n. This one can be written as sigma over i, sigma over j 1 to m A i times indicator function of A i intersection B j. So, S 1 can be written as this and we can write S 2 as sigma over i sigma over j 1 to m of B j times the indicator function of A i intersection B j. Now, union of A i B j, intersection B j, j equal to 1 to m, union i equal to 1 to n is a partition of X. So, now, they have common partitions and when you say S 1 is bigger than S 2 means what? So, let us take a point X. So, if X belongs to X, then it belongs to 1 of A i intersection B j, S 1 has the value A i and S 2 has the value B j. That means S 1 of X, which is A i must be bigger than or equal to S 2 of X, which is B j on A i intersection B j. So, that means if this is the representation, then S 1 bigger than S 2 implies that A i is bigger than or equal to B j if X belongs to A i intersection B j. Once you observe that, now problem is solved. So, what is integral of S 1 D mu? That by definition is sigma over i 1 to n, sigma over j 1 to m of A i mu of A i intersection B j and A i is bigger than B j if X belongs to this. So, this is bigger than or equal to sigma i equal to 1 to n, sigma j equal to 1 to m of B j mu of A i intersection B j and which is equal to integral of S 2 D mu. So, integral of S 1 is bigger than integral of S 2 if S 1 is bigger than or equal to S 2. So, that proves the next property. Now, we are going to look at special functions. If S 1 and S 2 are non-negative simple measurable functions, then we want to look at S 1 V S 2 and S 1 wedge S 2. Recall, how was S 1 V S 2 defined? S 1 V S 2 was defined as the maximum of S 1 and S 2 and similarly, S 1 wedge S 2 was defined as the minimum of S 1 and S 2. So, when we had shown that if S 1 and S 2 are non-negative simple measurable functions, then the maximum of S 1 and S 2 and the minimum of S 1 and S 2 are also non-negative simple measurable functions. So, we want to check this property now here that integral of S 1 wedge S 2 is less than S i integral less than or equal to integral of the maximum. But, that is obvious because if S 1 and S 2 are simple measurable non-negative simple measurable functions and you look at S 1 wedge S 2 that is the minimum of S 1 and S 2, then clearly S 1 wedge S 2 is the minimum. So, it is going to be less than or equal to S 1 and also going to be less than or equal to S 2 and S 1 V S 2, the maximum is going to be bigger than S 1 and S 2 both. So, it is going to be less than or equal to S 1 maximum S 2. So, what we are saying is S 1 wedge S 2 is less than or equal to both S 1 and S 2 and both S 1 and S 2 are less than or equal to maximum of S 1 and S 2 and just now and all are simple functions. So, what we have proved just now? So, that will say that the integral of S 1 S 2 the minimum of S 1 and S 2 d mu is less than or integral of S 1 also less than integral of S 2. So, less than or equal to integral S i d mu i equal to 1 and 1 and 2 and both these integrals are less than or equal to integral of S 1 wedge S 2 d mu. So, that proves the required property and that follows from the earlier property on that if S 1 is less than or equal to bigger than or equal to S 2, then integral S 1 is bigger than or equal to integral S 2 and now let us look at a property. How does this integral behave with respect to limiting operations? So, we want to claim that if S n is a sequence increasing sequence in L 0 plus. So, it is a increasing sequence of non-negative measurable functions increasing to a simple function S of X then integral S d mu is limit n going to infinity integral S n d mu. So, this is the first in a sense non-trivial argument required. So, S 1 S n are functions in L 0 plus non-negative simple measurable functions S n is increasing to S, S belonging to L plus 0 non-negative simple measurable. We want to show this implies that integral S d mu is equal to limit n going to infinity of integral S n d mu. So, this is what we want to show. Now, let us start observing. So, first what is the proof of this? So, note what we are given is S n is increasing to S. So, that means what? If S n is increasing to S that means that S n of X is going to be less than S of X for every X belonging to X. So, that is obvious from this. If this implies S n is increasing to S implies each S n X is less than or equal to S of X. Now, S n is a simple function. S is a simple non-negative simple measurable function. S n is less than or equal to this for every n. So, that implies that integral of S n d mu is less than or equal to integral S d mu for every n. So, integral S n d mu is less than or equal to integral S d mu and integral S n d mu is an increasing sequence of non-negative extended real numbers. So, implies that the limit of that which exists may be equal to plus infinity. S n d mu is also less than or equal to integral S d mu. So, here is that a n is a sequence of non-negative extended real numbers. a n less than or equal to a implies a n's are increasing. So, limit of a n will be less than or equal to a. We are in extended real numbers. Keep in mind. So, we have proved. So, let us call it as 1. So, we have proved in the required equality we have proved that right hand side limit n going to infinity integral S n d mu is bigger than or equal to integral S d mu. We want to prove the other way round inequality also. So, to do that, here is a small manipulation that we have to do. So, for that what we do is the following. .. Let a number c between 0 and 1 be fixed. Then c times S of x for any point x, c times S of x is going to be strictly less than S of x. So, here is c times S of x and here is S of x. So, let us fix c between 0 and 1 and look at the n for any point x. Let us look at c times S of x. Then the first observation because c is between 0 and 1, c is strictly less than 1. So, c times S of x will be less than S of x. So, it will be somewhere here. Now, S n x is increasing to S of x. So, after some stage S n x must be on the right side of c times S of x. So, after some stage it must be on the right side of. So, this is the picture that will happen. So, let us define b n to be the set of all x such that S n of x is bigger than c times S of x. So, collect all those points where this is going to happen, where S n of x is bigger than. So, this stage will depend upon n. So, that means, implies that first of all, let us note that b n plus, if S n x is bigger than c of S n, then S n plus 1 is anyway bigger than S n of x because S n is increasing. So, S n plus 1, x is going to be bigger than. So, b n. So, that means, this b n is inside b n plus 1 for every n. That means, that is b n is an increasing sequence. So, implies b n is an increasing sequence. So, that is the first observation because all b n's, S n is increasing. So, if x belongs to b n, then S n x is bigger than c times S of x. But S n is increasing. So, S n plus 1 x is going to be bigger than S n of x. So, if S n x is bigger than c times S of x, then S n plus 1 also is going to be bigger. So, x belonging to b n implies x belongs to b n plus 1. That means, b n is a subset of b n plus 1. That means, b n is an increasing sequence of sets. And also observe that each b n is an element in the sigma algebra S. Each b n is an element in the sigma algebra S because b n is where S n is bigger than c times S. All are simple measurable functions. And we have observed that such sets are in the sigma algebra. So, b n is an increasing sequence of sets in the sigma algebra S. Let us observe what is the union of these b n's. So, union of b n's n equal to 1 to infinity. Obviously, it is contained in x because all are subsets of x. But by the fact that for every x, picture that we observed here, for every x there is going to be some stage after which S n is going to be bigger than x because c times S x is strictly less than this. So, that fact implies that this union is equal to x. So, observation here is because for every x belonging to x, there is a stage n naught such that S n naught x is bigger than c times S of x. That is because S n x is converging to S of x because S n x is going to increase to S of x. So, it has to cross over this point c times S of x. Otherwise, it cannot reach that point. So, b n is an increasing sequence of sets in the sigma algebra and their union is equal to x and mu is a measure countable additive. We had proved a equivalent way of saying that mu countable additive is same as saying whenever a sequence of sets a n is increasing, then mu of a n's must increase to mu of a. So, by that fact mu of x must be equal to limit n going to infinity mu of b n's. So, that must be true. So now, let us use all these facts and look at. So now, so thus if we look at integral of c times integral of c times S of x d mu x, you look at this integral of c times S of x d mu x over b n's. So, first of all we claim that this is equal to integral of c times S of x d mu x over b n's. So, first observation we want to make at that and that is because if we look at this as a measure, if we look this as a measure mu of b n, just now we proved that integral over sets of simple functions over sets is a measure. So, look at that measure mu and b n is increasing to x. So, mu b n must go to mu of x. So, this fact we are using for this is the fact we are using for not mu, but we are using for mu and where what is mu? mu is integral of c S x over b n. So, that is the fact we are using here. So, that means this is equal to, so now on b n, what is happening on the set b n? On b n, S n of x is bigger than c times. So, that means c times S of x is less than, so it is less than integral over b n of S n of x d mu x because that is the definition of the set b n. So, we are replacing c S x, we are using the fact integral of S 1 is less than integral of S 2 whenever S 1 is less than or equal to integral of, whenever S 1 is less than or equal to S 2. So, this is less than or equal to this. Now, b n is a set subset of x. So, this integral I can replace and say that this is less than, so this is less than or this is less than or equal to integral over the whole space x S n x d mu x. So, what we are saying is by this analysis, what we have shown is that the integral c times S of x d mu x is less than or equal to this for every n and because this happens for every n and S n, these integrals are increasing sequence of numbers. So, this implies that integral c times S of x d mu x is also less than or equal to integral over x of S limit. So, it is less than or equal to limit n going to infinity of integral S n d mu and now this holds for every c between 0 and 1. So, I can take the limit as c goes to 1. So, implies that integral of S d mu is also less than or equal to integral limit, less than or equal to limit n going to infinity of integral S n d mu. So, that is my other way round inequality 2. So, we have proved both ways inequalities 1 and 2. So, 1 if you recall we had already shown 1 that integral S n d mu is less than integral S d mu that was 1 we proved and now we have proved integral S d mu. So, 1 plus 2 implies that integral S d mu is equal to limit n going to infinity integral S n d mu. So, that proves the result the required result namely that integral of S n if S n is increasing sequence in L plus then you can interchange. So, what is S that is a limit. So, integral of the limit is equal to limit of the integrals whenever S n is increasing non negative simple functions. So, it is a nice property for increasing sequences. So, at this stage one can ask the question that we have proved that if S n is a increasing sequence of non negative simple measurable functions increasing to S then integral of S n converts to integral of S. Will this property hold for decreasing sequences namely if S n is decreasing non negative simple functions decreasing to S can we say that integral of S n will decrease to integral S. We do not know that fact at present we cannot prove at present this fact. In fact many more properties of such things will explore as we extend the notion of integral. So, we will stop here today and analyze next time another way of representing integral of non negative simple measurable functions and then go over to define integral of non negative measurable functions. We will extend the notion of integral from non negative simple measurable functions to non negative measurable functions. We will do it next lecture. Thank you.