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00:09:420Paolo Guiotto: So… I'd like to start with, some exercises.
00:18:850Paolo Guiotto: from Chapter 4, these, exercises on, on, integral with respect to an abstract measure.
00:29:510Paolo Guiotto: So… Exercise, 4, 3, 4.
00:36:880Paolo Guiotto: We have a measure space.
00:44:820Paolo Guiotto: We have a measurable function.
00:54:590Paolo Guiotto: on E with the finite measure, So, E has positive And finally, measure.
01:05:170Paolo Guiotto: and F is positive and bounded, so F is greater or equal than zero, less or equal than a constant. Well, actually, it is not always through this, but…
01:18:50Paolo Guiotto: every X in E.
01:23:510Paolo Guiotto: Now, it says that…
01:26:100Paolo Guiotto: If you know that the integral f d mu is equal to M times the measure of E, necessarily F must be equal to M almost everywhere.
01:47:00Paolo Guiotto: So, let's start to understand, why…
01:53:960Paolo Guiotto: It is reasonable piece, because after you complete, actually, for that man.
01:59:440Paolo Guiotto: So I have most of the other issues with the function, the calculation. So if you place N here, so you complete the job of n, you have the figure that data of the constant is constant times the measure.
02:14:30Paolo Guiotto: So this quantity, with this assumption, is always bound by this one. They cannot be detailed.
02:21:440Paolo Guiotto: Now, it says that it gives equality.
02:27:410Paolo Guiotto: The necessary F must be constantly close to M, almost everywhere. So let's say that it cannot be less than M on a positive measure set. This can be… also, we can…
02:41:220Paolo Guiotto: recast the problem. So, let's see the solution.
02:46:240Paolo Guiotto: So, the thesis consists in proving that, well.
02:52:10Paolo Guiotto: something equal, something true almost everywhere means that that property falls on a measure zero set. So, the measure…
03:03:440Paolo Guiotto: of the set where F is different from M,
03:07:330Paolo Guiotto: is 0. And since we know that F is between 0 and m, this is equivalent to say that the measure of the set where F is strictly less than m is equal to 0, okay? Because it is less or equal. So this is the thesis, what we have to do.
03:27:290Paolo Guiotto: So… Let's see what happens if the thesis is false.
03:34:420Paolo Guiotto: So, if the function is smaller than M on a positive measure set. So, assume…
03:44:550Paolo Guiotto: I would start in this way, that the measure of the set where F is strictly less than M is positive.
03:53:480Paolo Guiotto: Now, we have, to try, of course, to get a conservation.
03:59:20Paolo Guiotto: And in particular, I would expect that if the function is on a positive measure set, smaller than that M, that integral cannot be equal to m measured. This is the goal, no? So, the goal now is, let's take the integral at that time that must be simply smaller than that value, so we have a confirmation.
04:22:40Paolo Guiotto: Now, this… so…
04:26:680Paolo Guiotto: Let's see if we can get that from this, the integral on e of f, d mu becomes smaller than mu dB. If this happens, we automatically have the conclusion, no?
04:40:320Paolo Guiotto: Because this would be a contribution.
04:42:750Paolo Guiotto: So now the problem is, can we say that?
04:48:610Paolo Guiotto: Well, an actual thing here we could do is I take the integral.
04:55:460Paolo Guiotto: And we split, since we have this information, on the set where F is less than M and the set where F is equal to M. I could say F less than M
05:05:710Paolo Guiotto: And here I have my F emu.
05:09:640Paolo Guiotto: plus integral, well, F is equal to M, NDIFF.
05:17:290Paolo Guiotto: Now, what is the problem with this argument? Because now, if I could say that this is… this is the, say, the delicate passage.
05:27:340Paolo Guiotto: That this is less… Since F here is less, strictly less than M,
05:34:70Paolo Guiotto: what if I increase, strictly increase the function set of integral? Can I say that this is strictly less than the integral on the same set of m d mu?
05:46:250Paolo Guiotto: Well, that property in general is not true, okay?
05:50:520Paolo Guiotto: So, when you increase F, you can only say that it will be less or equal.
05:57:330Paolo Guiotto: Okay, if I could say this, let's say for a moment that this is correct, I would have that, of course, this is equal to M, because there, I'm integrating function on the set where the function is M.
06:12:340Paolo Guiotto: So I would add here, n times the measure where f is less than n, plus
06:18:930Paolo Guiotto: M times the measure where F is equal to M, and therefore, this is M
06:27:40Paolo Guiotto: You know, these two sets are disjointed, because either F is simply less or equal, no? So they are this jominitor, this is the sum of two measures, it becomes the measure of the unit.
06:39:920Paolo Guiotto: No, but I think it's the other contrary.
06:42:790Paolo Guiotto: less or equal than M, which is the full set E. So I will get that this is the measure of the set where F is less or equal than M, but that's the domain. That's E.
06:58:20Paolo Guiotto: So I would get the contradiction, provided this strict sign is true, no?
07:03:710Paolo Guiotto: Now, since we have to make a formal discussion, we have to make sure that if this is correct or not, you see.
07:13:600Paolo Guiotto: Awful.
07:15:560Paolo Guiotto: Now, to be sure that this is correct.
07:19:700Paolo Guiotto: I… I will do this argument by taking a number which is smaller than M.
07:24:710Paolo Guiotto: Okay, so in that sense, I can… I can put a lesser equal, but it will become strictly less. So what I'm doing is the following. Since I know that the measure where we are supposing that the measure where F is strictly less than M is positive.
07:44:620Paolo Guiotto: Can I say that if I, take a little bit smaller number than M, this is still true?
07:53:710Paolo Guiotto: So, for example, can I say that from this, it follows that the measure of the square F is less or equal… taking less…
08:04:620Paolo Guiotto: You guys?
08:06:310Paolo Guiotto: That M minus something positive.
08:11:550Paolo Guiotto: Steven?
08:14:130Paolo Guiotto: some epsilon positive.
08:17:940Paolo Guiotto: Why?
08:21:760Paolo Guiotto: Because if this is correct, we can repeat the previous argument, but now it becomes easier. Because, let's take, let's call it star. If star is true.
08:36:570Paolo Guiotto: we can, without any questioning about this sign, no, strict or weak sign, we can say that the integral on e of f can be splitted into the integral f is less than m minus epsilon of F,
08:53:110Paolo Guiotto: plus the integral where f is greater or equal than m minus epsilon.
08:58:290Paolo Guiotto: of that.
09:01:750Paolo Guiotto: I want to show you that this quantity at right is strictly less than M times the measure of the set E, no? Which is supposed to be equal to the integral. Now, here, I can say that this is less or equal
09:18:300Paolo Guiotto: I don't need to care about if it is a strict or strong sign, yeah?
09:24:360Paolo Guiotto: M minus epsilon integrated on that set where F is less than M minus epsilon in the measure mu.
09:34:530Paolo Guiotto: plus the same here. Now, you put outside this constant, you get m minus epsilon, now you are integrating 1, so this comes the measure of the set where F is less than m minus epsilon.
09:50:320Paolo Guiotto: Plus, here, remind that F is always less or equal than M, no?
09:57:30Paolo Guiotto: So if I, if I put M here, I increase the function, so I will increase the integral. So you have M,
10:06:670Paolo Guiotto: times the measure where F is greater or equal than M minus epsil.
10:12:760Paolo Guiotto: Now, you see that if I factorize that M, I have the measure where F is less than M minus epsilon, plus the measure where F is greater or equal than M minus epsilon.
10:26:300Paolo Guiotto: And these two together, they are disjoint, because you cannot be in both, and their union is whatever, no? And whatever means the domain E. So this is the measure of E. But then, you see that there is this minus epsilon times this quantity.
10:44:440Paolo Guiotto: And this, this,
10:47:150Paolo Guiotto: Let's emphasize here, minus epsilon measure of set where f is less than N minus epsilon.
10:54:960Paolo Guiotto: And if epsilon is positive, and this quantity, this measure is positive, this number is a strictly positive number that I am subtracting
11:05:420Paolo Guiotto: to N measure of E. So, I have N measure of E minus something positive. So, this would be less, would be less than measure of E.
11:18:10Paolo Guiotto: So, M measure of P, but this is now a contradiction, because we started from this guy here.
11:26:360Paolo Guiotto: which is supposed to be, by assumption of this problem, equal to M measure of E now, and we proved that it is actually strictly less than M measure of E.
11:39:430Paolo Guiotto: So that's impossible, and this means that the initial point must be false.
11:45:70Paolo Guiotto: And so we have the conclusion. Okay, so… We would have a contradiction.
11:55:810Paolo Guiotto: And then we would have the conclusion. Now, the point is, okay.
12:00:610Paolo Guiotto: We said, by contradiction, we assume that this measure here
12:07:130Paolo Guiotto: is positive, no? The measure of the set where F is less than M is positive.
12:12:930Paolo Guiotto: Now, can I say that if I diminish that constant, and they take a constant which is slightly smaller than M,
12:21:890Paolo Guiotto: Hmm? Can I still say that that measure is positive?
12:31:950Paolo Guiotto: So let's, let's, let's write down here, you think, huh? So, we are assuming
12:41:810Paolo Guiotto: the proof of staff. We are assuming…
12:46:340Paolo Guiotto: that the measure of the set where F is frequently less than M is positive.
12:52:580Paolo Guiotto: So the question is, can… we… Say… And if yes, why?
13:00:720Paolo Guiotto: that,
13:03:890Paolo Guiotto: I don't know, you can… you can… you don't need to introduce the epsilon, you can use another constant, which is smaller than n, okay? But whatever, let's…
13:13:210Paolo Guiotto: to use that epsilon. Can we say that there exists a positive epsilon such that the measure of set where F is less than m minus epsilon is still positive?
13:26:80Paolo Guiotto: Can we say this?
13:28:490Paolo Guiotto: Of course, the argument here is based on the fact that we can say that is true. But how?
13:38:680Paolo Guiotto: Yes, that comes from the continuity, no? Because the idea is, look at the set where F is less or equal, less than M minus epsilon.
13:51:00Paolo Guiotto: What is the relation between this set and that set, where F is less than N?
13:58:240Paolo Guiotto: Well, if you are less than M minus epsilon, you are definitely less than M. So, the relation between these two guys is this one.
14:09:270Paolo Guiotto: Okay? But at the left, we have a positive epsilon, so we can imagine that there are extremely many veggies for this epsilon, and they can be as small as black, so this constant can be made…
14:23:660Paolo Guiotto: Very, very, very close to this one.
14:27:640Paolo Guiotto: Now, we have at left a family of sets and had tried one set. So, we would attempt now to take what?
14:38:240Paolo Guiotto: Well, the limit set at left, you may also realize that if you decrease epsilon, imagine you reduce epsilon, what happens here?
14:52:610Paolo Guiotto: So by increasing epsilon, you increase the constant basically below N. So these sets are increasing, you see? This is an increasing family of sets. So this… the natural operation would be take the union of these sets.
15:10:580Paolo Guiotto: for all possible positive epsilon. Well, if we write in this way, this is not a countable unit, but we can always discretize this. For example, we could take epsilon equal 1 over n, and then is natural greater or equal than 1, and we have
15:27:200Paolo Guiotto: a discrete version of this unit, so that's not the problem.
15:30:860Paolo Guiotto: Now, this unit here, let's think about, is made of sets which are contained here, so it is definitely contained into that set, so it is contained into the set where F is strictly less than M.
15:48:270Paolo Guiotto: But it also contains the set.
15:52:90Paolo Guiotto: No? Because if you pick an X inside this set, okay, you have a specific X, a specific value f of x, a number which is less than n, no? So if you fix an X,
16:06:610Paolo Guiotto: There, you have M, the value F of X will be somewhere here.
16:12:800Paolo Guiotto: At left, at strict left, if you pick a volume here.
16:17:190Paolo Guiotto: But then you will find an epsilon small in alpha, such that n minus epsilon is here, no? So pick an epsilon small in such a way that M minus epsilon is here.
16:30:240Paolo Guiotto: And this figure says that if X belongs to this set, to f of x is less than n, f of x will be less than n minus epsilon, which means X belongs to one of them, or X is not sufficiently motor for some X.
16:48:430Paolo Guiotto: So it means that if you are in the right set, you are in this unit, so that the inclusion becomes a double inclusion, and so it is
16:57:800Paolo Guiotto: inequality.
16:59:610Paolo Guiotto: And moreover, we realized that this is an increasing value, so we can use the continuity.
17:09:250Paolo Guiotto: And, the family of sets where F is less than n minus epsilon.
17:16:40Paolo Guiotto: is an increasing family of sets, so this means that the limit
17:21:560Paolo Guiotto: when epsilon goes to zero, I write with epsilon, but you understand we have to discretize, so, I don't know, n going to infinity, and epsilon is 1 over n. Nothing changed. Of the measures of these sets.
17:36:570Paolo Guiotto: This is equal to the measure of the set where F is less than n, which is supposed to be positive.
17:43:650Paolo Guiotto: Now, if the limit is positive.
17:46:860Paolo Guiotto: at least some of the elements must be positive, otherwise the quantities, well, here cannot be negative. If they are all equal to zero, then it would be zero, okay? It would not be positive.
18:01:670Paolo Guiotto: So this is saved from this force I mentioned, some of this form is strictly positive.
18:11:330Paolo Guiotto: So… the measure of F less than M minus F, you know?
18:16:550Paolo Guiotto: Strictly positive for… some.
18:20:360Paolo Guiotto: Actually, if you want to say more… If it is true that for some action, this is positive, for infinity, many axion is positive, because for the axiom is smaller than that one, the second is bigger, many should be greater. For some one, it's positively, for the other, it's positive.
18:38:90Paolo Guiotto: There are infinitely many of these sets for which we summit. Okay, so this we… with this, we close the argument completely.
18:48:60Paolo Guiotto: Okay, so I've shown you this exercise because you have… For example, D435…
19:00:420Paolo Guiotto: It's different, but it's based on the same idea, okay? So, if you have not done the 434 yet, now review this and try to do the 43.
19:12:130Paolo Guiotto: 35…
19:17:380Paolo Guiotto: So let's do…
19:19:290Paolo Guiotto: another one, which, the 439, I think there is some problem with the text, because,
19:26:240Paolo Guiotto: there is an epsilon where it shouldn't be, so, not look at that, maybe I will…
19:33:520Paolo Guiotto: the exercise. Let's do the 4-3-8.
19:40:590Paolo Guiotto: This guy says that we have an F, which is in a 1 of a certain space, X, with this property, such that
19:52:940Paolo Guiotto: the integral on e of f with respect to measure mu is positive, or 0, that sense.
20:00:140Paolo Guiotto: for every set, measurable set.
20:05:920Paolo Guiotto: So, I'm saying you have a function which is equal, and you know that all integrals, no matter what is the integration domain, are false. What would you expect on the function?
20:19:260Paolo Guiotto: When these happened, sir?
20:21:530Paolo Guiotto: It's not true.
20:25:370Paolo Guiotto: This problem happens if F is positive, no? So, if F is positive, this is automatic.
20:34:540Paolo Guiotto: Now, what the exercise has to prove is that, actually, if this happens, F must be… Then, F…
20:44:00Paolo Guiotto: must be greater or equal than zero. Of course, almost everywhere.
20:49:820Paolo Guiotto: No? You reminded the vanishing theorem, no? We add that the integral of a positive function equals zero in some domain, the function is zero. Not necessarily everywhere, but almost everywhere. That's happened with integrals. We'll never be able to have properties that are everywhere true.
21:09:900Paolo Guiotto: Okay? When they… they are… When the measure of the integrity is involved.
21:17:450Paolo Guiotto: Okay, so how could we proceed here?
21:26:400Paolo Guiotto: Did you try this?
21:28:900Paolo Guiotto: No.
21:30:180Paolo Guiotto: Please do exercises.
21:32:620Paolo Guiotto: It is fundamental, it is much more important than studying theorems. Well, you have to study the theory, because maybe you have some kind of idea on how to handle these things.
21:47:700Paolo Guiotto: Okay, so, I would say, let's assume the conclusion is false. So, what does it mean that the conclusion is false? If false.
22:01:120Paolo Guiotto: It means that it is non-true that F is equal, greater or equal than zero almost everywhere. So this means, by the way, let's write, this means that the measure
22:12:190Paolo Guiotto: Of the set of points for which this property is false, This means F-negative.
22:19:200Paolo Guiotto: So the measure of the set where F is negative is 0. This means the conclusion, no?
22:25:390Paolo Guiotto: So, if false means that that measure is positive.
22:31:200Paolo Guiotto: If also a measure of add negative, Must be positive.
22:40:300Paolo Guiotto: Now, the idea could be, okay, let's take this as E,
22:45:900Paolo Guiotto: Because if we integrate such F on that E, what could be the expected?
22:51:980Paolo Guiotto: Integrals.
22:54:290Paolo Guiotto: Now, we are integrating on that domain a negative function, something which is negative.
23:00:920Paolo Guiotto: So, clearly, the integral will be less or equal than zero.
23:05:330Paolo Guiotto: But not sure if it is strictly less.
23:09:640Paolo Guiotto: If it is less than 0, it can be equal to zero, so it is not in contradiction with that property. But if I prove that it is strictly less than zero, it is in contradiction.
23:23:300Paolo Guiotto: all should be, can we say that? I told you, we are using what kind of product? We will not only see the update, we increase the function, we increase the integral, but this is never a strict, property, okay?
23:37:740Paolo Guiotto: So…
23:38:870Paolo Guiotto: About, issues, state exemptions, etc. But let's say that, in general, we can have only weak, so less than equal inequality. Then, how could we, handle this to get strictly less?
23:58:510Paolo Guiotto: So, the problem is that, here, I get natural equivalent because I'm mapping this by zero.
24:04:500Paolo Guiotto: So I get 0 times the maximum equals 0.
24:09:600Paolo Guiotto: But if I count down this by a negative number, this becomes for sure negative.
24:16:270Paolo Guiotto: So, you say, similar to the previous exercise, how can I do that?
24:22:190Paolo Guiotto: So, that there is an epsilon positive such that the measure of where F is less than minus epsilon, so this is a negative number, is positive.
24:37:920Paolo Guiotto: It sounds fair, okay?
24:40:530Paolo Guiotto: So, yes.
24:44:650Paolo Guiotto: as… in, previous… Problem.
24:51:230Paolo Guiotto: So, basically, the same type of argument. You can see that the set where F is negative is a limit set of sets where F is less than minus epsilon with epsilon positive.
25:05:300Paolo Guiotto: is when you send epsilon to zero, that minus epsilon goes to zero, and you approach F less than zero, so…
25:13:470Paolo Guiotto: So if we can do that, we can say that then the integral on that E
25:20:530Paolo Guiotto: Which is the set where F is less than minus epsilon.
25:26:690Paolo Guiotto: of the function f, by assumption must be greater or equal than 0. This is the assumption.
25:35:590Paolo Guiotto: On the other hand, since F on that set is less than minus epsilon, you can say that this is less than or equal than the integral on the same set of minus epsilon
25:47:920Paolo Guiotto: the measure mu.
25:50:190Paolo Guiotto: So, I increase the function by replacing F with the constant minus epsilon. And this is equal to minus epsilon
26:01:220Paolo Guiotto: minus epsilon, the measure of this, F less than minus epsilon.
26:07:380Paolo Guiotto: But now, as you can see, this number we have here
26:12:40Paolo Guiotto: is strictly positive, so with the minus in front, becomes strictly negative. Why? Because epsilon is supposed to be positive, so it's non-zero. And we have selected this epsilon in such a way that this measure
26:28:240Paolo Guiotto: is positive. So this number is strictly greater than zero, so we can say that the minus that is strictly less than zero, and here you see the contradictions. Now, 0 is strictly less than zero.
26:52:970Paolo Guiotto: And therefore, this means that The, contradiction assumption.
26:58:210Paolo Guiotto: Must be false, and therefore the conclusion must be true.
27:02:440Paolo Guiotto: Okay.
27:06:200Paolo Guiotto: As I told you yesterday, we will not insist too much on the LeBague measure, because most of time we will work with one variable integrals for our applications, and so we just need to…
27:23:110Paolo Guiotto: Confused, if it is the case, yeah.
27:30:720Paolo Guiotto: Yeah, because probably, so… The point is that,
27:39:290Paolo Guiotto: I am recording, but since the network is not working, I'm using my mobile, and
27:48:900Paolo Guiotto: So let's do that. Probably, I've opened another meeting, so let's do that.
27:54:380Paolo Guiotto: Let's stop this one.