Hello, good morning traders. How are we all doing today? If you could let me know, sound level is good. Any echoes today? You can use the hashtag advanced traders techs channel. Great. Okay, so welcome to the the live advanced webinar here at book map. Today we're joined by Jay Trader. We do this every Wednesday around 10 a.m. Eastern Time. Now the sharp traders amongst you will have noticed something a little bit different this week. Bruce is away on holiday. Probably is currently sipping cocktails in the Caribbean somewhere. So my name is Sam and I'll be assisting Jay Trader today. Bruce fans, don't worry. He'll be back next week fully refreshed probably with a nice tan. So you can read more about Joseph Joseph's trading career here on his website jtrader.co and connect with him on on Twitter and YouTube Jay Trader Co. I've just got some disclosures to quickly read. So bear with me before we start. So the general disclosure all book map limited materials information and presentations offer educational purposes only and should not be considered specific investment advice nor recommendations. Live trading is in simulation demo paper trading mode and strictly for educational purposes. Live trading executed in simulation cannot accurately represent realistic trading performance and the risk disclosure. Trading futures equities and digital currencies involve substantial risk of loss and is not suitable for all investors and Investor could potentially lose all or more than the initial investment. Risk capital is money that can be lost without jeopardizing one's financial security nor lifestyle. Only risk capital should be used for trading and only those with sufficient risk capital should consider trading. Past performance is not necessarily indicative of future results. Okay, with that done, Joseph, I believe you're here already. Yeah, how are we doing? Yes, thank you, buddy. Glad to have you here today. So we'll start the screen. Let me see the screen over here. Yep, I'll let you take it away. We're actually distant neighbors, Joseph. I'm down in Malta at the moment. Are you in San Marino? Yes, where are you in Malta? Just near Valletta, just south. Oh, okay. Yeah, I know, I know. It's a very nice place. Is it worn by you then? Yeah, no, the sun is out today. Yes, it's the sun is shining. It's a history place. It's a history place. Yeah, for sure. Plenty of history here. Yeah, beautiful place. So can you see my monitor, Sam? Yes, I see you. Okay, perfect. So good morning, everybody. If you see the monitor, just put over here. Yes, if you see it, good. All right, so the market is unwinding this morning. And I say right away what I traded. I posted before OISDK and DO. And I had a plan over here to wait, first of all, for direction ES and NQ. ES, we simply started to dump, right? So don't look at all you see over here, but focus more only traders on the red bars over here and on this Aqua band, right? So you can see that the Aqua band shifted below these two other bands over here, the Mazen to the yellow. And it's the bars, these bars are red and it's simply fading, fading, fading, fading, fading. Okay. So yes, the main guy. Personally, I didn't take trades, as I said, on the big caps today. Only I traded OISDK. I considered that the mid cap NVIDIA faded, but only around 945 or 950. We had Apple, so Apple also unwinding. We had Netflix. So all the tech stocks faded, right? The one that faded the most is Tesla. So look over here at Tesla. This is, first of all, what the market is doing on Tesla. We're still above. This main support has low, the support around 792. And on the long term, you can see that the stock is still up training, right? So this is really the moment, the next weeks will be the moment that I believe we decide the market direction. It's not going to go up or down in NVIDIA, Apple, Facebook, Amazon, Google. Everything depends from the overall market. And right now we have really Bitcoin, that even if it's not a tech stock, because we're talking about a different technology, and we're talking about a cryptocurrency, right? It's moving very close to how Tesla and other tech coins, other tech stocks, sorry, are doing. So this morning, I'm looking over here for a possible short on Tesla. I'm not looking for any kind of a divide. The support over here, you can see 796 is basically this, over here, this major level of support. And then you can see the reclaimed 800, so a whole round of number. So I will look for more jail and rejection as a spot over here to short. Or if you're looking to scalp, there was a lot of, there were a lot of opportunities over here to trade with the bear flag. So over here, and then we add over here, and you can simply write this down all the way. Again, don't short trade it because I see a lot of traders trying to short too early or they short too late. You have to have a perfect timing. In the meantime, INDO is pushing. And this is a small cap. We're talking about 1.5 million market cap. And I'm looking to short this. So I'm looking for a short around 82830. This is a stock that I traded this morning over here. There we are. Eight plus, they're trying to push it. And right away, they stuffed it. You can see they're selling over here, right away, some sweeping indicator printing the dots. So when we have a push right away, a fast move, you will see that sweeping indicator. So now let's wait. This morning, I traded exactly this year over here. I posted before my trades. Two times I shorted around 810 and 830 the second time. Then I took the partial covers into 780. Now you can see that Z-Line is how it works, right? We held it over here and then pushing. Also traders, if you see other extension plays, please alert. So I'm looking again for 820, 830 to short, short on stuff. And this is the previous level of before. And what I'm looking also over here, you can spot my level 830, right? This level over here of high liquidity, which is this era. And then we have a little bit more 840, where was the previous high 840 over here. So I'm using basically my levels on the chart together with the levels over here on Bookmap. Hope that makes sense. I started short over here, 820. So I took the top tick, 820 over here, got the short, but only a nibble, okay? I didn't load the boat over here yet. You can see they're trying to push it over here, volume above half a million. I will show my execution once done. But you can see trader, this is high volatility over here, big sweep. So pushed down over here to the 785. So I'm covering over here partial, looking for the first covers over here in the 770. And why I'm saying that? Because I've seen a fast move over here right away, slam dunk, okay? So you can see over here how they sold it. Look on the times and sales, right? You can look over here on the times and sales, use amount of sellers and perfect trade over here. So we're looking for that fail. Now I don't see any kind of big buy over here should fail more to the J-Line of 760. Again, I didn't fully load it. Below 770, yes. Now look traders over here, main support around 760, okay? So this is the first real main target. We trap longs, I mean, not personally we, I mean, it's figurative over here. Main area of, I would say, selling pressure, second main area of selling pressure. But you can see that over here, we had a push to 840. And then over here, lower volume, a push over here only to 820. So possibly over here, dump and boom, washed to 750 and ideally 730. So looking for that price formation, look over here, the tape, 766, 770 you're selling, 4,000, 4,000, 5,000. Here we are. J-Lines over here, 760. Want to see a breakdown, 760. And then I want to see a 720 and then a 7, all right? So remember, this is not the easiest stock in the world to trade. It's not the easiest. We are over here just below the open level. Why isn't it easiest? Because for example, we have a lot of volatility. We have a lot of spikes, micro float. So if you want to short the stuff, you have to be very careful. So I trade big caps, crypto, futures and small caps. But I like this stock because yes, they had a lot of volume. And when I see rains, there's a lot of possibility to make a lot of money. So that's the reason why I'm trading INDO, right? If I want to see something more simple, I just look at, for example, here on Tesla. And I start looking for those patterns over here. You can see this red line, start looking when it fades like this. You look to short the pops and then use book map as your confirmation. So when you have a pullback, you start looking over here for a pair of flag and then you're looking for that short. Okay, hope it makes sense. Some support over here, the 795, you can see they're stacking the support. 795, 790, 785. So we have some big buyers looking at the stock. This is an important support. As we said before, this 790, 795 and looking at the stock over here, we have that possible, we have a possibility over here for breakdown. We have to look for this 790 breakdown for more white. Looking at my position INDO, I covered over here partial, I'm still waiting. This is what did yesterday. So you can see how big of a move this was, went from something like 480, 470 until nine and then double top over here and then faded. This is using the one minute chart. So you can see it's time goes the spikes, these levels over here, you see these bands then fades down, pop. In this case, we didn't pop to 840, which was my favorite level too short, but it eventually came over here. Three stocks, I was looking today INDO or STK Tesla. I traded this morning or STK, I traded a pop and filled the gate. I shared this in the room. I took two points with the half size INDO, possible breakdown coming over here. So look this micro support. They formed this micro support. Somebody over here wants to get filled, but I don't see any kind of big buyer. 765 goes away, washed to 750, 765, possible breakdown incoming. 55 went. Now 62. They're trying to dump over here more. You can see only sellers on the tape. In this case, when it's in tape means times and sales. And this day was a cool chart. So I believe this was a form of why I don't see it anymore. Cancel this. Okay. So this I wanted to show you. So when this morning I was trading this with a few traders in my room, I was looking for a fill. So I started to short over here. You can see that we don't have any more liquidity. Now I didn't put over here today the dots because I don't want to focus much on that. I want to focus more on the levels. Right? So every time I try to focus for you guys on something different. In this case, you see that each time we went to the VWAP, so this red line, we had over here wash and right away backup bounce, wash bounce, wash bounce. Okay. So this is the sweep indicator that the other time we were looking with Bruce. What I like over here is that if you look below, we don't have any kind of main support, right? No support, small over here, small over here, but then they simply cancel, right? So they go away from the from the book. While over here we have seller, seller, seller, more seller over here becomes even becomes even bigger, more aggressive, more seller, more seller, more seller, right? So the first time we reject over here, it basically doesn't have a lot of resistance, but above we have some wall of sellers between the eight and the 35. Okay. So this is only 30 cents. And then we go exactly to push into this resistance at eight, sorry. So basically what happens over here, we have a first seller over here and then I believe that a him cancels to jump in over here and wants to get filled limit order.  All right. So these are jumping in or I believe one of these two over here probably canceled from here got to here over to here. So they want to really get failed traders. And then the boom, there we are 740 and that's the boom. And that's the boom. So now looking for the seven, there we are. Boom. Perfect magic. I'm not really looking to cover yet. We have no support. So for me, there's no point I'm covering this. Okay. No reason to cover INDO. We're gonna look for INDO over here, five minute jay lines. In this case, we're looking for seven. You can see no support below. Nothing over here that I, you know, is showing me any potential bounce. 720 big wash over here. I'm going to take partial. There we are. I'm going to take partial around six, nine years, seven. Most traders today, another beautiful trade which was Quinn GI and just saw it late and I love this extension over here. These are pumping dumps with very low volume. You can see the volume over here was 270 K. So nothing, right? Wow. It's totally different from something like INDO yesterday. That for example, when I had this extension had volume over here of nine and 101 million. All right. So total difference scenario. Now what you have to look for is can I still see over here any kind of buyers on INDO? So where it bounce? Okay. That's what you have to look at. And not really. I'm not really seeing any kind of bounce over here, any kind of buyers. I see more seller. You can see over here, everybody wants to get out. Now they're afraid. Now they're afraid this morning. They probably bought in over here because they thought it was going to do something like yesterday. Now this is fade from the top one buck per share. So that's a huge fail over here. Huge fail over here. So 720. Possibly another rejection coming. You can see over here. They're building some support, right? Some resistance. Look at this. And this is really what you have to look when you're looking at trail because it's very easy to enter a trade. You're short or you long, whatever, let's say we long over here would have been the same thing. But then you have to be able to recognize if the trend is still going in your favor. If the trade is still working, if you're making lower highs lower lows or in this case, it's a short, of course, if this is continuing to downside, you have to be super, super, super attentive in this. It's one of the most important thing because a lot of traders think it's only about the execution of entry. No. This is not really too, this is just to show you and to motivate you. I just want to show something over here. And there's a reason why I'm showing this. So this is a trader that I'm coaching and he started over here the first two months and then went to 40K. This is basically four months of work together. You can see from December, from October, sorry. And you can see his win rate is 46%, right? This is Jordan. So 46% is a not high win rate. But the fact that his average return on winners is much bigger than average return on loser, that's the reason why he is making money. So you don't need really to have a 90% win rate. Absolutely, you don't need that. You need only to have, in this case, a high return low risk ratio. So we're talking about R. Because in this case, the trailing and the use of the stop loss makes a difference. Okay, so this I wanted to explain you traders. This is super important when you trade. Okay, super important. I'm watching over here that now we don't have any more than the resistance, but we have a very fluff support. When I'm looking for a fader, I'm looking for this 90 may. Okay, it's just the normal 90 may. A lot of times you will see this breeze to the upside, and then right away, they slam down. So if I see the same scenario that we saw over here, you can see it's going down, it's popping right away. Boom. Somebody slamming over here is pushing down the bid. Okay, selling over here. Then the same thing could happen right here. That means stay in, stay in, stay in. Let's go back and check Tesla. In the meantime, so still holding that support. So this is Tesla. The same main support 795 and then we have 790. Let's check this on a five minute chart. And for now, even here, you can see a lot of sellers. So I want to use the same chart I'm using on INDO. So you do like this, right click, charge settings, inherit charge settings from, okay. Then each time it happens, you do that, we'll come up over here, the CVD. So just close it if you don't need it. I'm not, as I said, focusing today on the dots. I'm mostly focusing over here purely on the price section and on the levels. I want to check again, Tesla on the one minute chart. And we're going to look at a few things together. But as you know, my plan starts with the one minute chart. And then I refer to book map and to the five minutes are as confirmation. I'm going to take off this. I'm going to put over here inside bar. So I want to show you something that it might be useful for you. Inside bar display. We can make this every color we want. We can make, for example, this blue or pink, all right. See if blue works, our colors. We're having traders basically a volume divergence over here. Higher lows, higher highs, sorry. Let's see. Actually, no, not volume divergence. No, not yet. So where I want to trade over here at Tesla. These bear flags or J-Lines pops. Nothing else. For now is a lateral over here. There's consolidation. It's normal when you have these faders, you can see in the morning. Okay. Just look at the easy stuff. Just look at this perfect J-Line rejection. They're trying over here to hold this 800. You can see a lot of orders out over here. Seemed like they're spoofing, right? First, they come over here. So first they come over here. Then they move down here. Okay. So while the price is going down, orders over here are moving. Then this is pushing up and this is moving over here. Then in this case, this is coming down. He's not getting executed over here. It moves. So look right now. We're talking about that the rejection 90 May is rejecting. I'm not scalping Tesla. Okay. So I'm not really looking to trade these pop and fail over here. Again, a double bottom on triple bottom on this support. Let's check again, traders INDO over here. So they're trying to hold that. You can see before we had no buyers. And now look when a stock wants to reverse what they do. Start forming over here, support. Support exactly into this main J-Line. And then over here again, over here, and over here, and over here, and over here. Okay. So in this case, you can see traders broke the 730 down over here to the 725. No buyers, right? What does it mean in a fader like this with no buyers that can easily fail? Now I'm looking only over here on the right. And the last big orders we had were, yeah, the majority were on selling on the cell over here. 719 INDO. Look what a wash over here. 711. Nice juicy wash. Still in that support 720. I'm waiting over here. So I'm having my position, simply waiting. 729. A little bit of buyers you can see over here. Look over here, they're curling, right? So they're putting some buying over here. And now came a 724 and then 730 around 6,000. Over here, look traders. Let's go to Tesla. Still in that support. Again, I'm having over here a toss, but this is a broker that you cannot use for small caps. I'm only using for charting over here. I'm looking the fill that we're having with Tesla. So finally, it's breaking that support 795. Yeah, sure. Yeah, I've just been looking for the money. Okay, let's look also D-Wax. So I'm not looking to buy Tesla over here. Just a lot of patience. Okay, we need to wait over here. D-Wax. Let's see Facebook, Facebook, I'm winding as well. These days, I'm trading mRNA and Vax. I mean, mRNA has a beautiful fader on the daily. This is hourly chart, but you can see each time we go to the J-Lines, basically, it fills, right? So it's one of those stocks that you can look simply to short these levels over here for a swing or each pop. It's a pretty good swing over here on mRNA. Let's go back to Tesla. So main support 795 holding over here, bouncing. And until we don't get 807, I'm not looking to touch this. That's my main point right now. Starting to bounce. Again, you can see each time this small over here is scrolling, right? Just below that 800. So careful when you break it at 100 because it can push. Am I'm longing over here? No, not really countertrend. So no edge really trading right now. We have to wait. No edge rejection from the 800. Again down. So attacking that support over and over and over. This is something very nice on Tesla. The fact that we're having a multiple over here, we say scenarios, right? That we're attacking the support and this is a main daily support. So if you're looking at trading Tesla, you can see that the last two weeks over here, we were lateral. Again, when we have this environment of basically possible work, it's called crisis. Market always reacts bad. Even if then there won't be a war. Why? Because market suffers the indecision, suffering the uncertainty. And what happens? Generally, market tends to fail, pull back to the downside. Therefore, the tech stocks are the ones that are starting to fall in the first place. Why? Because they're also the ones that when the market is training strong, the one that gave you best range. And market generally tends to increase the price of oil because you know the exportations. For example, right now, there was a deal that came off with Russia and Germany. Also Europe, almost half Europe, gets oil and gas from Russia. So of course, it will be like an increase of the price of oil. And what that means, crude oil will go up, investors will buy ETF on oil, and then the market generally tends to go down when we have this kind of scenario. Now, right now we have something I would say not totally new because still Bitcoin was, you know, since 2010 over here, we could trade it. But we have to see if Bitcoin will follow as they're doing right now the tech stocks. Tech stocks are falling, Bitcoin is falling. And we saw this more than once. All right. So there are also other, I would say, factors that influence the price of Bitcoin. I explained this, I made a video about it, so go watch it, it's free. But right now, let's go back to the market. Market is really trying to show us over here a direction, but we still cannot say we have a direction. Just look at the S&Q, the last one month and a half. Okay. Then the big caps, if you take Tesla, Apple, Nvidia, Bob, Amazon, Google, Facebook, you will see that they gave maybe two or three days up, like you can see over here, then they are like fading, then up, then fading, then lateral and fading. So it's not a market that I really love to trade unless I'm day trading and unless I'm scalping. All right. Yes, they were the big downtrend, but it's still not a bearish market. Why? Because on the long term, we're still having this push. And even in the past, we have some pullback into this area here. So into the J-Line one, J-Line two, same over here. Okay. So if you break 790, then we can have a really big unwind to the 790 on Tesla, but this will depend from many factors. Of course, the index will be in premise the leading guide if you trade tech stocks, right? So I hope that traders is clear for everybody. So what to do in this market? I'm shorting or longing small caps, and you'll tell me, J, how to do it. Okay. First of all, do a forecast in the morning, make your analysis, back test, journal, takes a lot of time. And this is something like INDO, all right? So we see over here that yesterday I was looking to short this, and I took a small cat over here in the morning. And then over here, we had this curl for a big push. You can see over here, big push, then a stuff. And then this is an A plus setup too long. This is an A plus setup. So you don't really need to buy over here and then sell at this top. Okay. It's impossible. I will never buy over here and hold and sell at the top. Who tells you this doesn't know really what he's doing. Or it's just like smoke. You want to buy over here once we have the fake break done. And right away over here, you can see this push break the previous high, and then you can start selling in partial. So what I do is trailing, selling in partial. When we are up here, I raise my trail. When we are up here, I raise my trail. When we are up here, I raise my trail. And eventually, when we fail, I'm getting out. So you'll never get the perfect top. Maybe if you have a pivot, if you have a previous resistance, you will take a partial, but not you will buy entirely over here, you will sell over here. But that doesn't really, that is not really important. You invest over here something like 6000 bucks. And this goes up to eight. If you already make an average over here and you sell at seven, that's 1000 bucks. All right. So with 6000 investment is not a bad return on something like this. What is around 18% something like that. And then you can see over here that you can still buy these dips, but each time we have over here lower highs, lower lows. So this is like a wedge. When you have a wedge, be careful because you can have or a fast push or right away a fast breakdown. And then what they do over here, this is the first, we'll say mid shelf over here breakdown, where they sell over here, partials.  And over here, they trap a little bit more until like they're satisfied. And you can see over here, we go up again, we go test this previous high, and then boom rejection. This over here is the best setup that you can do if you want to long. All right. But then you can have even others over here. And today, for example, we had the same stock, but I simply nailed a short and killed it today. Short in this level, but I took only the majority till here and then in the money. And then I reshorter over here. Unfortunately, not full size, but I still took a decent profit, decent profit, and then I'm looking for a possible one of here. So a stock like this will give you a ton of, a ton of, I would say return if you can trade it good. The other one that we had this morning was OSTK. So we have earnings, but earnings were not good. So now I show you how the market works, right? So let's put over here this. So they basically did this. Look, the company had over here a strategic feed zero investment. So it's a strategic collaboration, whatever. But we had something like an earning miss. Why? Because the active customers fell of 12% in the four quarter and the orders were down 25%. Okay. So despite this, OSTK went up. And this morning, my plan I shared on YouTube before starting, I said, I'm looking to show this area. I got feel 48 and something with few guys covered over here. And then I didn't trade this anymore. And then over here, if you want to really trade this easy, you show it over here. So this is OSTK. In this case, you can say, wow, Jay, they made a collaboration, strategic agreement, whatever. It's a beautiful news for the company. Okay. You have to think that it's already absorbed over here in the gap. So it already went from 40 until 50. It's more than $1 billion company. We're not talking about small float. And this is really a lot. Went up 20% over here or more. Okay. And then I do one thing that I suggest you do. So when you join to the room over here, but soon this will be available for everybody on the subscription for now, it's free. I'm going to check over here, the stats. Okay. So these are the stats. This app is called beta as to trade. As to trade is called. It's a better version. Team is uploading over here, I would say some stuff today. So it's not really uploading. But I can tell you that I saw this morning, gap stats OSTK, when we have a gap above 20% and at least 5 million volume trade on the day right now is 22 million. Okay. So gap up 20% at least of gap, 5 million volume at least trade on the day. We had two times this in the last two years. And we had two times this faded and closed below the open. What is doing right now? Morning push closing below the open because these news over here. So strategic collaboration partnership are basically fluff. Okay. Are basically, you know, sell the new scenarios program. And you don't want to long and hold something like this. Make sense traders. Just, you know, put a yes if it's clear or not. Okay. Thank you, Charo. I think it's maybe only clear for you. Okay, gondola. Thank you, buddy. It's very important to understand how the market moves. The price action, the, we say the big hands, the whales. Okay. This is really the stuff that you need to learn. Thank you, Dom. Thank you, buddy. Let's go back over here to Tesla. Still holding that support. Actually, sorry, we're below that support, but the daily chart. But here we can put this as a daily chart. The low, this is around 791. So let's move this to 791. There we are. Okay. That's the main support. Just selling over here, selling, selling, selling. 791. So we are at that main level. I'm going to wind over here on Tesla. I'm still short over here at 9DO. We need to break this low. Average is 808 and another broker, 811. So this setup over here, traders that I'm watching is still telling me to go for a short. Still holding for short, ideally to recycle. In order for me to recycle, I want to see a pop exactly into this area over here and then unwind. Still no movement over here. Let's see it back again, OSTK, which has a little bit of resistance, but again, we are not at that level. Before we had a general rejection at 1055, exactly into this area over here, this area over here, exactly here, 4550. I believe it's this over here. No, this over here. So this level, pretty, very good. Pop into the J-Lines, hit them out above. You can see over here, price goes to a resistance, then they build more resistance, right? You start seeing the lower high is this area over here. And I want a short each time that I have this blue area for an easy short setup. Feel like I don't have this level. I'm not looking to short it. Let's look over here. Again, at INDO, breakdown coming on INDO. Check this curl down over here, looking for coverage into the 670, 660, this area over here. Nice juice and wine. So you can see two things over here, the same. I'm going to explain over here. So we have INDO, right? And just look when the red line goes below, and the same thing for the long side, right? The red line over here goes below. This is INDO. It's below the ACCO one. Over here, you have the same, right? Same exact thing. You can see over here, it goes down and then you have the pop. Basically, two same setups right now. 69 over here, INDO, beautiful. This is unwinding good. You stay in right now. I want to get that fader. I want to get that fader. Okay, Trader. So we have the last few minutes together. Any questions, first of all? Something you want to see, you want to go over with? Every morning, you can follow us on YouTube where I share a book map over here in the trades, the pre-market plan. Then every Wednesday, we are over here with this case, today's Sam, other times with Bruce. Anytime we check something different, this today, we cover more, for example, the small cap INDO. Previous weeks, we cover more Facebook and other big caps. It really depends what the market offers us. 90. Again, no buyers over here. Look at that tape. That's really the last stocks. So Facebook breaking the support as well. We have another support at 199.200 and then more fader. We call it 200 over here like weeks ago. And if you follow us, OSTK, the five-minute jailings we fill, we're going to go with 40. Let's see, Apple. Next target, 162. NVIDIA, one of the stocks that I always trade, support at 230 and then support at 225. So we still have some possible unwind over here. Let's see, Netflix. Netflix at the lows, 372, 373. All right, Trader. So I think we cover some good stuff. Sam, are you still here, buddy? Any questions, Sara, Gondola, Dom, anybody else? Hello, Joseph, you hear me okay? Yes. Oh, good. Yes, I have a question for you. I liked your trade management description. That was great. Maybe you could expand a little bit on that.  Are you always looking for a specific R? Do you use trailing stops? Yeah. Good question. So trailing is what, first of all, gives me, I would say, an account with a low win rate, but I still become profitable through trailing because trailing increases my R return. So whenever I start trading, I first of all find my targets and my stop loss area. What I mean by that, let's say that today we traded, I don't know, let's put a one minute short, let's say that today we traded Apple. Before I even take the trade, I calculate, it has to be sometimes so quick, really a glance, I calculate my risk-reward ratio. So a different setup, of course, different risk management. In this case, let's say I'm looking for a short, so I will be looking for a trend line break. This is going to be my R, okay? So to calculate your risk-reward ratio, what do you have to do? This area, till this area, which is my stop, is calculate in one R. This one R could be $2, $1, $0.50, whatever. So when I go down, if I know that my win rate, let's say it's 45%, I'm going to look for at least two R, maybe even three R of return, if I want to be highly positive and highly profitable. So in that case, I start looking, okay, where is the possible three R now? Just like doing by looking over here with eyes, this would be like $2.50, this maybe is like three R return. So I know that ideally, my average of return has to be this one. So I don't have to enter and skidding out, skidding out, skidding out. So by the time I get over here, my R return is only one R, okay? So that's the first important thing, calculating the Rs. Then you have to look out the trail. So for this, you need order of flow. So as we were doing before, you're still looking on the book map traders, if you see over here in times and sales, sellers. So if you see red or if you see green, for example, and how many big support I see below. And once we get those supports, are we really bouncing or we still find the huge amount of sellers that want to dump and make the stock unwind? And then each time I form a lower high, I start putting my trailing. So when I break one, I can trail partial and then I can hold the remaining with the jailers over here, because it's still going down, it's based on my back testing on my journal, my tracking. This is a very good tool to use. Okay, so different scenarios, different patterns, of course, different ways to trail. But that's like, for today, I believe the trailing method that I'm using on for the majority on the big caps when I trade. Makes sense, Sam. Yeah, no, thank you. That's great. It's good to see how you kind of calculate it in terms of R. Because I think babysitting the position is one of the hardest things really. A lot of people focus on the entry, but kind of maximizing your return and staying in the position is the tricky part. Exactly. Because even often what happens, maybe I take two or three trades, only a nibble, only a feeler, and maybe those two, three trades are not going to work. So I'm going to stop them. Then when I start seeing instead that something is working, and I have possibility, for example, to load the boat to add to my position, to trade around the core, those are the trade that will give me the paycheck. And maybe these trades will be generally 20, 30% of the entire amount of stocks that I trade or whatever that I trade. So traders don't enter that position, maybe it's wrong and you start adding, adding, adding, you know, maybe dump size, cut it, then you can reenter. If you take even one, two, three cuts, and then maybe you have one that really works, don't worry, you will get paid. Yeah. No, and it really makes the point that it's actually very challenging to trade with one lot. If you're a futures trader, or just one contract, it's hard to scale out. It almost becomes impossible. You got to get the correct position. Yeah, you have basically to pinpoint and, you know, it's almost impossible because to have the best, I would say, risk-worth-worth ratio, because you have to basically, you know, get out at least at the first main target, but then you could have a second, a third, a fourth, nobody knows, you know, how much, for example, can unwind if you're short or run if you're long. So that's the reason why I, in that case, I would suggest maybe trading micro contracts, maybe like 10 or 20, or even whatever you want, but at least you can scale out instead of trading, you know, only one future contract. At least you have to trade two contracts for the futures, at least. Yep. No, awesome. That's great. Thank you. All right. So thank you, Sam, for being a great host over here. Thank you, everybody. Charu, Gondola, Dom, Barbosa, Ryan, OG, Sam, Jimmy. Thank you, everybody. If you need help, reach out to me, Bruce, or Sam. We are here to help you. Hopefully, you learned something today that you can reuse in the next days and study it. Thank you very much, traders, and see you next week. Thank you, Sam. Thank you, everybody. Thank you, Joseph. Take care.


The Adaptive Card Online Designer also supports previewing the card in the various hosts as I showed you earlier. And previewing the card and testing out the interaction in various hosts is pretty useful during the development process. So again, you do that by clicking the preview mode button in the top navigation. Notice the card structure and the element properties panels they change when the designer is in preview mode. Now I'm gonna select the due date that you see listed here to see how it's gonna behave when it's rendered in the host app. So here I select due date, we can see how it's changing around.


Welcome to lecture number 13 on measure and integration. If you recall in the previous lecture, we had been looking at the Lebesgue measure, Lebesgue measurable sets and its properties. We will continue that study of Lebesgue measurable sets and its properties today itself. We will be especially looking at the translation in the various properties of the Lebesgue measure and then Lebesgue measurable sets vis-a-vis topologically nice subsets of the real line. Let us recall that we had defined what is called the Lebesgue measurable sets and that gave us the space, the real line, the Lebesgue measurable sets and the Lebesgue measure. This was called the Lebesgue measure space and the Borel sigma algebra of real line, the Borel subsets of the real line form a sub sigma algebra of Lebesgue measurable sets. This properties we had seen and now today what we are going to look at is the following. Recall that on the real line, there is a binary operation of addition. You can add real numbers. This operation can be used to transform subsets of the real line. Let us take a set E contained in real line and define what is called E plus x. E plus x is defined as all elements y plus x such that y belongs to E. It is the set E which is translated by an element x. The question is if E is Lebesgue measurable, does this imply that E plus x is Lebesgue measurable? Similarly, we will also look at the second question namely if E belongs to Br, if E is a Borel subset of real line, does it imply E plus x belongs to Br? These are the two questions we will start analyzing. The importance of these two questions is the class of Lebesgue measurable sets. Is it invariant under translations and is the class of Borel subsets invariant under the group operation of translation on the real line? These two questions we will answer in the first to start with. To answer the first question, let us recall that Lebesgue measure is nothing but the restriction of the outer Lebesgue measure. The Lebesgue outer measure for real line is defined as the infremum of sigma lambda of intervals i i, where these intervals form a covering of E is a subset of union i i's i equal to 1 to infinity and i i's are all intervals. Keep in mind the remark, we said you can choose these intervals i i's i j's to be open if necessary. Whether you take all possible coverings of E by intervals or all possible coverings of E by open intervals, both will give you the same value namely the Lebesgue measure of the Lebesgue outer measure of the set E. Let us start with a set E, which is Lebesgue measurable to show that E plus x is also Lebesgue measurable. This is the question. To show that, recall what is the definition of a measurable set? To show that for every subset y of real line, we should have Lebesgue outer measure of y is equal to Lebesgue outer measure of y intersection E plus x plus Lebesgue outer measure of y intersection E complement plus x, E plus x complement. That is what we have to show. Let us start observing that we are given that E is Lebesgue measurable. So, E Lebesgue measurable implies for every subset y of real line, the Lebesgue measure of y is equal to outer Lebesgue measure of y intersection E plus outer Lebesgue measure of y intersection E complement. Now our aim is to transform this E to E plus x. That means we should be looking at the properties of the outer Lebesgue measure of a set in terms of translation. Let us observe. Let us note that if a set E is covered by a union of intervals i j 1 to infinity, that is if and only if E plus x is covered by union of the translated intervals i j plus x j equal to 1 to infinity. That means every covering of the set E gives a corresponding covering of the set E plus x by the intervals i j plus x. Note that if i j is an interval i j plus x also is an interval. So, if E is covered by intervals i j, we get a corresponding covering of E plus x by the intervals i j plus x. Conversely, given a covering of E plus x, we can construct back a covering of E by looking at by translating by minus x. So, this property obviously implies that Lebesgue measure of a set E is same as the Lebesgue measure of the set E plus x. So, this observation implies that the Lebesgue measure of a set remains invariant under translations. So, this is a property. So, this is an important property of the Lebesgue outer measure we are going to use to conclude that if E is Lebesgue measurable then E plus x also is a Lebesgue measurable. So, as we said E Lebesgue measurable implies for every subset y, we have got lambda star of y is equal to lambda star of y intersection E plus lambda star of y intersection E complement. And now observing that Lebesgue measure is invariant, outer Lebesgue measure is invariant under translations, we can write this as lambda star of y intersection E plus x plus lambda star of y intersection E complement plus x. So, here we are using the fact that lambda star is translation invariance. And now, a simple observation I will tell you that y intersection E plus x is same as y intersection, it is same as y intersection y plus x intersection E plus x. So, meaning that if you take intersection and translate, it is same as translating and intersectionist. So, that is simple set theoretic property. So, the first term here is equal to lambda star of y plus x intersection E plus x. And similarly, the second one will give you it is lambda star of y plus x intersection E complement plus x. So, that is using the fact that lambda star is translation invariant and intersection translation is same as translation intersection, they commute with each other. And now, this property is true for every subset of y. So, I can replace y by y minus x. So, implies replace y by y minus x, we get this is also equal to. So, lambda, this property is true for every. So, let us replace lambda star of y minus x is equal to lambda star of y plus x minus x. So, that is y intersection E plus x plus lambda star of y plus x minus x. So, that is y intersection E complement plus x. So, in the above equation, we replace y by the set y by y minus x. So, this is true. And now, observe lambda star of y minus x is same as lambda star of y by translating y or y x or minus x does not affect. So, we get for every subset y, lambda star of y is equal to lambda star of y intersection E plus x plus lambda star of y intersection E complement plus x. Now, a simple observation tells you that this set E complement plus x is same as E plus x complement. So, first take the complement and then translate that is same as saying first translate and then take complement. So, it is purely a simple set theoretic exercise, which you should be able to verify easily. So, we get lambda star of y is equal to lambda star of the first y intersection E plus x plus lambda star of y intersection E plus x complement. Hence, this implies that E plus x is Lebesgue measurable. So, we have proved the first property namely if you take a Lebesgue measurable set E and translate, then the translated set also is Lebesgue measurable. So, another way of saying the same thing is that the Lebesgue measurable sets are translation invariant. They remain the class of Lebesgue measurable sets is translation invariant. And we have already seen the Lebesgue outer measure is translation invariant. So, that means that the length function is translation invariant on the class of all Lebesgue measurable sets. So, this proves the first property. So, we have answered the question that if E is Lebesgue measurable, then E plus x is also measurable. Let us look at the next question that if E is a Borel set, can we say that E plus x also is a Borel set. So, to answer this question, we need some topological properties of the real line. So, let us look at the topological properties, what we are going to look at. So, the question is if E is a Borel subset of real line, does that imply E plus x is also a Borel subset of real line? So, for that let us observe consider the map. So, consider the map from real line to real line, where y goes to y plus x for every y belonging to y. So, x is fixed. So, this is translation. So, this is the translation map from real line to real line and the observation is that this map translation is a homeomorphism. So, what does homeomorphism means? That is it is 1 1 on 2 and both ways continuous. That means this is continuous and the inverse map because it is 1 1 on 2 that also is continuous. So, this is an important property very basic, but yet important property that we are going to use to prove that if E is a Borel set, then E plus x is a Borel set. So, to prove our requirement, let us consider the collection say S. So, S is a collection of all subsets which are Borel and which have that required property namely E plus x belongs to B R. So, look at all Borel subsets of real line which such that the translation translated set is also a Borel set. So, we are going to prove two things about this. One that the class of all open sets all open sets are subsets of S and second thing will prove that S is a sigma algebra. So, once these two properties are proved all open sets are inside S and S is a sigma algebra. So, these two properties will imply that the sigma algebra generated by the class of all open sets is inside S and that is nothing but the Borel sigma algebra. So, the Borel sigma algebra will come inside S and that will prove the required property. So, we have to only prove these two facts namely that if you take an open set then it belongs to S. That means if I take an open set and translate that should be a Borel set. But that is again obvious by the fact that translation is a homeomorphism. So, if E contained in R is open is an open set then the set E plus X is also an open set. So, that is also an open set. In fact, that is a very simple property to prove because if you take a if E is open that at every point there is a open interval inside E and the translated one will be inside E plus X. So, that is easy to verify or one can simply verify by saying that E plus X is an open map or the translation is a homeomorphism. So, this is an open set. So, this implies the first property that all open sets are subsets of S. The second thing that it is a sigma algebra. So, for that note empty set the whole space are both open. So, they belong to S and second if a set E belongs to S that implies E plus X belongs to S is a Borel set. But that implies this is a sigma algebra. So, E plus X complement is also a Borel set because it is a sigma algebra. So, must be closed under complements and now a simple observation that this set E plus X is nothing but E plus X complement is same as E complement plus X. So, this is same as this. So, that belongs to B R and that implies that E complement belongs to E complement belongs to S that is same as saying E complement. So, the class S of subsets of Borel subsets of real lines either translates or Borel sets includes empty set the whole space. It is closed under complements and let us prove that this is also closed under countable unions. So, third property namely, so let E n be a sequence of sets in S implies that E n plus X is a Borel set in R and Borel sets being a sigma algebra that implies union of E n plus X also belongs to B R. Hence, now here is a simple observation that this set is same as you first take the union and then take the translation. It is same as first translating and then taking the unions. So, that belongs to B R. So, basically translation commutes with all set theoretic operations that is observation we have been using again and again. So, this belongs to B R so that implies union n equal to 1 to infinity E n is a set in S. So, we have proved so hence open sets are inside S a sigma algebra. So, implies that the Borel sigma algebra generated by open sets which is the Borel sigma algebra is inside the smallest sigma algebra generated by open sets namely the Borel sigma algebra must also come inside S and that is a subset of B R. So, hence all are equal. So, this proves the second fact namely if E is a Borel set then it is translated translation is also a Borel set. Once again it emphasizes the use of the technique that we had called as the sigma algebra technique namely we wanted to prove that for every Borel set E the translation is a Borel set. So, note we collected all the sets which have this property and we proved two facts namely the open sets are inside this collection S and S is a sigma algebra. So, that implied that the smallest sigma algebra generated by open sets which is nothing but the Borel sigma algebra also comes inside S. So, this technique we will be using we have been using and will be using quite often is called the sigma algebra technique. So, you want to prove some property about subsets of a set X collect them together and try to show that those sets form a that collection forms a sigma algebra and includes the generators of the required sigma algebra. So, we have proved that under translation the measurable sets the collection of Borel sets are very well behaved and we get a translation invariant measure on real line that is the length function. So, this gives us the fact that lambda the length function on B R the Borel sigma algebra of subsets of real line is the unique translation invariant measure such that the length of the interval 0 comma 1 here comma is missing. So, 0 comma 1 is equal to 1 and this is a very important property of the length function. So, if you observe on the real line there is a notion of addition and we just now pointed out that the group operation X comma Y goes to X plus Y is a continuous map and one can also easily check that X going to minus X that is the inverse and the group operation is also a continuous map that is summarized by saying that the real line is a topological group. So, on the real line there is a topological structure there is a metric there is a topology on the real line there is a group structure on the real line and the two behave very well with respect to each other saying that the group operations X comma Y goes to X plus Y and X goes to X inverse both are continuous maps. So, one says such a thing is called a topological group. So, the real line with addition and the usual metric the normal metric forms what is called a topological group and we have shown on this topological group there exists a translation invariant measure on the sigma algebra of subsets there exists a translation invariant measure. This is a very important fact and that can be generalized to what are called locally compact topological groups that on every locally compact topological group there exists a invariant measure because the group may not be abelian. So, one has to make a specific thing there exists a left invariant or a right invariant measure on every locally compact abelian group and that plays a important role in doing analysis on such groups. So, just a pointer that you may come across in your other courses in higher studies of higher mathematics that on every locally compact topological group there exists a invariant measure and that actually is called Haar measure on the topological group. So, Lebesgue measure on the real line is an example of Haar measure on the locally compact topological group real line under addition and the usual multiplication the usual metric space topology. So, these were the properties of Lebesgue measurable sets vis-a-vis group structure. Now, let us look at properties of the Lebesgue measurable sets with respect to topologically nigh subsets of the real line namely topologically nigh sets are open sets and closed sets. So, we will actually analyze and characterize measurability of sets in terms of open sets and closed sets. So, this is the precise theorem that we are going to prove namely if E is any subset of the real line then the following statements are equivalent saying that a set E is Lebesgue measurable is equivalent to saying that for every epsilon bigger than 0 there exists an open set G lower epsilon such that E is a subset of G epsilon that means that open set includes E and the difference between the two sets has got outer measure small. So, that means there is a little little difference between a Lebesgue measurable set and open set which covers it. So, we will show that for every a set E is Lebesgue measurable if and only if for every epsilon bigger than 0 there exists an open cover of it such that the difference between the cover and the set. So, lambda star of G epsilon minus E is small and we will show that this is equivalent to saying that there exists a G delta set G such that the set includes E and the difference has got measure 0. A set G delta set what is a G delta set? A G delta set is nothing but a intersection of open countable intersection of open sets. So, subsets of the real line or in any metric space which are countable intersections of open sets are called G delta sets. So, let us prove this theorem. So, saying that these three statements are equivalent is saying that if one of them holds then the other one also holds. So, what we are going to do is we will assume one and show that one implies two and then we will show that statement two if you assume statement two that implies statement three and if you assume statement three then that implies one and that will imply that all these statements are equivalent. So, if one of them is true then the other two statements are also true. So, that gives you a characterization of Lebesgue measurable sets in terms of open sets or G delta sets. So, let us prove this theorem. So, we will start with looking at. .. So, let us assume. So, let us suppose the statement one holds. So, that is E is Lebesgue measurable. To show two and that is for every epsilon bigger than 0 there exists a set G epsilon open such that G epsilon includes the set E and the Lebesgue outer measure of G epsilon minus the set E is less than epsilon. So, this is what we have to show. So, let us start with something this is regarding outer measure. So, let us start looking at the set E. So, let us first suppose E is such that E is Lebesgue measurable. So, let us suppose that Lebesgue measure of E is finite. So, what is Lebesgue measure of E? Recall, Lebesgue measure of E is same as its outer Lebesgue measure, which is same as infremum over sigma lambda of i j 1 to infinity where the set E is covered by union of i j's intervals and each i j open. So, recall we had made a observation that in the definition of Lebesgue outer measure, you can assume that all the intervals involved are open. So, let us fix and since this number is finite and it is infremum. So, by the definition of infremum there exists a covering. So, there exists intervals i j's such that E is contained in union of i j's, each i j open and we have got the property namely lambda star of E, which is same as lambda star of E, which is same as lambda of E because it is measurable plus the small number epsilon is bigger than sigma lambda of i j, j equal to 1 to infinity. So, and that implies, so this implies, so note lambda star of E is finite. So, this is finite quantity. So, that implies sigma lambda of i j, j equal to 1 to infinity is finite. So, all the sets I have got. So, this implies that if I take a look at lambda star of union i j, j equal to 1 to infinity, that will be less than or equal to summation j equal to 1 to infinity lambda of i j by the sub additive property of the length function and that is finite. So, this set union of i j's is a set of finite outer measures. So, let us define, so put G epsilon to be the set, which is union of i j's. Now, let us note that first of all G epsilon is open. Why it is an open set? Because each i j is an open interval. So, a countable union of open intervals is an open set is open and E is inside union of i j's. So, E is contained in G epsilon. So, we have got the required property that we have got a cover of E by an open set and let us note and what is the difference? So, lambda star of G epsilon minus E. Now, note G epsilon is an open set. So, it is a Borel set. So, it is a Lebesgue measurable set and E is given to be Lebesgue measurable and it is subset of it and everything is finite that we have just now observed. So, we can say this is equal to lambda star of G epsilon minus lambda star of E. So, here we are using the finite additivity property of the length function and lambda star of G epsilon that is here is that is same as lambda star of union i j because G epsilon is union i j minus lambda star of E and that is less than or equal to sigma by sub additive proper countable sub additive property 1 to infinity lambda star of i j minus lambda star of E and that by our choice of our intervals i j's if you recall this was the choice. So, that means lambda star of this summation minus this is less than epsilon. So, which is less than epsilon. So, lambda star of G minus G epsilon minus E is less than epsilon. So, that means what? That means we have proved the required property when lambda star of E is a finite set. So, thus if lambda star of E is finite then 1 implies 2. Let us remove this condition and that condition here is a step where is the important step which would keep in observation that we first proved a property about the length function for finite sets of finite measure and now we are going to extend this using the fact that lambda is sigma finite. So, whenever you want to one wants to prove a property about the length function or about sigma finite measures many a times it is easier to prove it when the underlying set is of finite measure and then extend it to sets general sets of sigma finite measure. So, that is what we are going to do now. So, let us in the general case. So, in general the set E which is Lebesgue measurable may not have a finite Lebesgue measure, but the Lebesgue measure being sigma finite we can write E as a disjoint union of sets E j, j equal to 1 to infinity such that lambda of E j is finite. It is a measurable set and it is finite. So, now by the earlier case because lambda star of E j is finite. So, for fix epsilon bigger than 0 there exists open set call it G j which includes E j such that lambda star of G j minus the set E j is less than the small number epsilon, but we are going to write less than epsilon to the power 2 to the power j and you will see soon why we are doing that because for each piece we are going to make it small and we are going to add up these pieces. So, now define the set G epsilon to be equal to union of G j, j equal to 1 to infinity. So, then G j is open. So, it is an open set because it is a countable union of open sets G j's and G epsilon includes G j includes E j. So, includes union of E j's which is equal to E. So, it is an open set which includes E. Now let us look at the difference further. So, let us observe further that G epsilon minus E what is that equal to that is union of G j's minus union of E j's that is the definition that is how we constructed. Now here it is a simple set theoretic property namely that this is a subset of union j equal to 1 to infinity G j minus E j. So, this is a simple set theoretic curve that union of G j's minus union of E j's is a subset of union of G j minus E j and once that is verified which is easy to verify we get that lambda star of G epsilon minus E is less than or equal to sigma j equal to 1 to infinity lambda star of G j minus E j which by our choice is less than epsilon to the power j. So, this is less than or equal to summation j equal to 1 to infinity epsilon by 2 to the power j which is equal to epsilon. So, that proves the second property completely in the general case also. So, hence what we have shown is that 1 implies 2 namely if E is Lebesgue measurable then I can find given epsilon we can find a open set G epsilon such that the outer measure of this is small is less than epsilon. So, now let us go to the second step of the verification. So, we have verified the first step namely 1 implies 2. Now, let us verify that 2 implies 3. So, let us assume 2 holds. So, 2 implies 3. So, we are given a set E. So, given 2 means let E be a subset of real line such that for every epsilon bigger than 0 there exists a open set G epsilon which includes E and lambda star of E is just a set. So, we cannot write lambda now it is lambda star of G epsilon minus E is less than epsilon. So, that is what is given to us and we have to construct a set say that the difference has got measure Lebesgue measure of 0. So, the obvious way is make this epsilon small and small. So, in particular so that says for every epsilon equal to 1 by n there exists G n including E G n open such that lambda star of G n minus E is less than 1 by n. So, we have specialized this given condition for each epsilon equal to 1 by n and we got an open set G n which includes E. Now, because we want it small s. So, we want to let this become smaller and smaller. So, it says the following. So, define G equal to intersection of G n n equal to 1 to infinity. So, what is G? G is intersection of open sets. So, note G is what is called a G delta set. So, G delta set is by definition an intersection of open sets and this set G includes E because each G n includes E. So, G also includes E. Further lambda star of G minus E note G minus E is a subset of G n minus E because G is the intersection. So, G minus E is a subset of G n minus E. So, by monotone property lambda star of G minus E is less than lambda star of G n minus E which is less than 1 by n. So, lambda star of G minus E is less than 1 by n for every n. So, that implies the fact that implies the fact that lambda star of G minus E is equal to 0. So, hence we have shown 2 implies 3. Now, let us conclude the proof by showing. So, let us show we show that 3 implies 1. So, what is 3? So, 3 says that by 3, so 3 implies that for E E a subset of R, there exists not an open set, there exists a G delta set G including E and lambda star of G minus E equal to 0. So, here is the set E that is the set E inside and this is the set G which covers it, so that the remaining part has got measure 0. But note what is E? E is same as you take the set G, this is the full set and intersect it with the complement of the outer portion. So, this is the complement. Look at the complement of this. So, intersection G minus E complement. So, this is a simple observation because what is G minus E complement? So, this inside portion is, so let me just draw a picture again. So, this is inside is E and outside is G. So, this shaded portion is G minus. So, what is complement? Complement is the outside portion here and E. So, when you intersect it with G you get E. So, E is nothing but G intersection G minus E complement and this is a G delta set and a G delta set is an intersection of open sets. So, this set belongs to B R and hence this set also belongs to L. So, it is Lebesgue measurable and this set G minus E has got outer measure 0. So, G minus E belongs to is a Lebesgue measurable set because all sets of outer measure 0 are Lebesgue measurable. So, this set also belongs to L and so intersection of two Lebesgue measurable sets is Lebesgue measurable. So, this implies that E belongs to L. So, 3 implies, so this implies 1. So, this proves completely the fact that the three properties that E is Lebesgue measurable is equivalent to saying for every epsilon bigger than 0, there is an open set G epsilon such that E is a subset of G epsilon and lambda star of G epsilon minus E. The difference is outer measure small and that is equivalent to saying that for the set E there exists a G delta set covering it such that the difference has got measure 0. So, this gives us a characterization of Lebesgue measurable sets in terms of open subsets of real line. A corresponding characterization of Lebesgue measurable sets is obtained in terms of closed sets. Let us state that also and prove it. So, let us look at the next that for any set E in R the following is true namely E is Lebesgue measurable is equivalent to saying for every epsilon bigger than 0. Now, there is a closed set inside E such that Lebesgue measure of the difference is small and that is equivalent to saying that there is a F sigma set. So, what is the F sigma set? A F sigma set is nothing but a set which can be expressed as a countable union of closed sets such that there is an F sigma set F inside E such that lambda star of E minus F is equal to 0. So, let us quickly prove this and this will use the earlier characterization. So, suppose one holds E belongs to Lebesgue measurable sets. Now, E belongs to Lebesgue measurable sets implies there is an open set which covers E with difference of measure small but we want closed sets. So, let us observe E belongs to L also implies that E complement belongs to L because this Lebesgue measurable sets is a sigma algebra. So, if E is Lebesgue measurable complement also measurable. So, this implies by just now what we proved. So, E complement is Lebesgue measurable. So, for every epsilon bigger than 0 there exists an open set G epsilon such that this includes E and outer Lebesgue measure of G epsilon minus E is less than epsilon. This is just now we proved this fact. So, this is of E complement. We are applying the previous just now proved result for E complement and now so if E complement is inside G that means E will be including G epsilon complement and note G epsilon is an open set. So, it is complement is a closed set. So, let us call it as C epsilon. So, C epsilon is closed. It includes E and we want to find what is the Lebesgue outer measure of E minus C epsilon. So, what is that? That is E intersection. What is this? This is just C epsilon complement by set theory and that is same as E intersection. This is G epsilon. So, we want E complement. We want to find out what is the difference between the closed set inside E complement. So, that is sorry not not not the complement E. So, E minus C epsilon is E intersection C epsilon complement and C epsilon complement is nothing but G epsilon G epsilon. So, this is G epsilon intersection E. So, we want to conclude. .. So, now note that Lebesgue outer measure of E complement. So, Lebesgue measure of we had E minus. So, what is this? So, this we can also write it as G epsilon minus E complement because E complement complement will be E. So, E minus C epsilon C epsilon is same as outer measure of G epsilon minus E complement and that is less than epsilon. So, what we have shown if E is Lebesgue measurable then there is a closed set inside it such that the measure outer measure is less than epsilon. So, implies two holds. So, one implies two we have proved and here we have just used the fact that a set is open if and only if it is complement is closed and applied the previous criteria. So, one implies two we have proved. Let us look at two implies three and that is once again for every epsilon what is given that there exists a closed set C epsilon closed inside E, C epsilon closed with the property that outer measure of E minus C epsilon is less than epsilon. So, in particular for every epsilon equal to 1 by n there exists a set C n which is inside E, C n closed such that Lebesgue measure of E minus C n is less than 1 by n. So, put so technique is same. So, put C equal to union of this sets C n. So, this 1 to infinity. So, this is a F sigma set because it is a union of closed countable union of closed sets and each C n is inside E. So, this set C is also inside E with the property that lambda star of E minus C. C is the union of all the C n. So, if I take only one of them C will be a subset of E minus C will be a subset of that. So, it is less than or equal to lambda star of E minus C n which is less than 1 by n. So, implying lambda star of E minus C is equal to 0 and that proves hence three holds. So, for given for every epsilon there is a closed set inside it we have shown there is a closed set inside E with difference there is a not a closed set a F sigma set with the measure 0. Finally, let us put that prove that 3 implies 1. So, E is a subset of real line E E is a subset of real line with the property that there exists a F sigma set C contained in E with the property that lambda star of E minus C is equal to 0. So, here is the set E and I have got a set C inside it. So, this is a set C inside it say that the difference has got measure 0, but now note the set E can be written as C union E minus C. So, this is this portion outside and union the inside portion and this is F sigma set. So, it is a Borel set and hence it belongs to is a Lebesgue measurable set the set E minus C is a set of measure 0. So, that is a Lebesgue measurable set. So, E is a union of two Lebesgue measurable sets. So, implies 1 that is E is Lebesgue measurable set. So, we have proved the third property namely 2 implies 3 and now we have just now proved 3 implies 1. So, the Lebesgue measurable sets are very nicely connected with topological nice sets namely open sets and closed sets. So, for every Lebesgue measurable set can be covered by an open set such that the difference has got Lebesgue measure small and which is equivalent to saying a set is Lebesgue measurable that is also a characterization and similarly a set E is Lebesgue measurable if and only if we can find a closed set inside it such that the difference has got measure small. So, Lebesgue measure is a very nice measure on the real line and it is defined on all Lebesgue measurable sets. So, in particular it is defined for all Borel sets and it is translation invariant it gives nice is compatible with the group structure and it also has nice properties with respect to the topological structure namely with respect to open sets and closed sets. So, with that we conclude our analysis of Lebesgue measure and Lebesgue measurable sets. In the next lecture we will start looking at functions on measurable spaces and their properties. So, thank you very much.