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00:16:79Paolo Guiotto: Okay, so… There's in the comments.
00:24:370Paolo Guiotto: So, as I… as I told before, it is, in general, it is difficult to define a sigma algebra. Normally, however, as you will see, it is not a big problem, because what we do normally is the following.
00:41:330Paolo Guiotto: We start from a family that we want to assign a measure, okay?
00:47:730Paolo Guiotto: We want to be sure that that family will have a measure. And then, since that family won't be, maybe, a sigma algebra, what we do is we find a way to extend that family to the cheapest possible sigma algebra.
01:04:800Paolo Guiotto: So we… we have a standard procedure to do that.
01:08:860Paolo Guiotto: And this is called the sigma algebra generated by a certain family. We give this result a proposition.
01:18:960Paolo Guiotto: So, let's, X B.
01:24:480Paolo Guiotto: Set…
01:26:650Paolo Guiotto: And let's say that we start from S, which is just a family of subsets of X, not necessarily a sigma algebra.
01:38:880Paolo Guiotto: We are finally… of… subsets.
01:48:110Paolo Guiotto: Off.
01:49:120Paolo Guiotto: Thanks.
01:50:540Paolo Guiotto: Not… necessarily, sigma object.
01:57:630Paolo Guiotto: So what is the sigma algebra generated by S?
02:01:730Paolo Guiotto: It will be a sigma algebra that contains XSS inside.
02:07:720Paolo Guiotto: And it will be also the smallest possible, so the cheapest sigma algebra.
02:14:470Paolo Guiotto: So let's say D.
02:18:570Paolo Guiotto: We call it Sigma Algebra.
02:21:870Paolo Guiotto: generated.
02:23:790Paolo Guiotto: Bye.
02:25:10Paolo Guiotto: S?
02:26:400Paolo Guiotto: We use this notation.
02:30:330Paolo Guiotto: We, we denote this by sigma of S.
02:38:300Paolo Guiotto: Ease.
02:39:600Paolo Guiotto: D.
02:42:210Paolo Guiotto: small list.
02:48:630Paolo Guiotto: Sigma. Algebra.
02:51:820Paolo Guiotto: Couldn't say anything, but…
02:57:340Paolo Guiotto: S.
02:59:30Paolo Guiotto: Now, how is defining this, huh?
03:01:780Paolo Guiotto: And it is given by this, let's say, formula.
03:05:970Paolo Guiotto: sigma of S.
03:08:450Paolo Guiotto: is what? You take all the possible sigma algebra that contains
03:15:70Paolo Guiotto: S. We will see that there are, okay?
03:19:520Paolo Guiotto: So we take the family of all sigma algebras, where F is a sigma algebra.
03:27:850Paolo Guiotto: And the F contains S.
03:32:10Paolo Guiotto: And we take the intersection of all of them.
03:37:670Paolo Guiotto: So, it's impossible that you have an intuition on this. Okay, we are creating families of sets.
03:44:820Paolo Guiotto: the unique condition is that they must be some algebras, and they must contain S inside. So it is clear that since each of them contains S, so that S is the script that.
04:01:270Paolo Guiotto: That's the script S. Since all of them contain S, also the intersection will contain S. So this thing, once it is defined.
04:14:880Paolo Guiotto: It contains X by definition.
04:17:750Paolo Guiotto: Now, the only problem is, is there at least one of these sigma algebras
04:24:470Paolo Guiotto: Is there certainly a sigma algebra that contains S?
04:30:30Paolo Guiotto: Whatever is S, whatever means X display.
04:35:770Paolo Guiotto: Just because you're reminded the trivial sigma algebras are two. One is the poor sigma algebra with just any index, and the other is the richest possible sigma algebra.
04:46:100Paolo Guiotto: Which is the parts of S.
04:48:740Paolo Guiotto: So, part of X is madra and contains whatever is the family of subsets, because S is contained in part of S. You see there? Part of X itself is a Sigmajra. So, clearly, there are sigmajras that contains S.
05:06:650Paolo Guiotto: at least one part of X.
05:09:400Paolo Guiotto: We don't know how many, maybe infinitely many, you can imagine how many, in general, if the set is infinite, big, etc.
05:19:100Paolo Guiotto: So, this explains why that definition is well posed. And it turns out that this is a sigma algebra.
05:28:870Paolo Guiotto: So, this… Sigma S.
05:35:240Paolo Guiotto: is well-defined.
05:40:870Paolo Guiotto: And that… It is,
05:45:110Paolo Guiotto: Well, just a remark on these, these two words that are often used in the mathematics. What does it mean, well-defined?
05:57:880Paolo Guiotto: Because I will often ask you, check that this thing is well-defined. Now, normally people have some difficulty in understanding what does it mean well-defined.
06:09:120Paolo Guiotto: Well, let's say that the ruler is the following. Whenever it's non-trivial, you have to check that it makes sense.
06:16:900Paolo Guiotto: For example, you compute an integral.
06:20:190Paolo Guiotto: But not every function is integral, alright? So you should check if the function is integral. You compute the derivative. Well, not every function has a derivative. You should check if it is differentiable.
06:32:490Paolo Guiotto: So, whenever you have something which is non-trivial, you have this… it smells that it is non-trivial, you have to check that it makes sense. So, the well-defined here, just to make clear, so proof.
06:47:600Paolo Guiotto: So, sigma S… is, well.
06:52:460Paolo Guiotto: defined. What does it mean here? Well, there is not a universal definition of what does it mean well-defined. It means that makes sense, okay? So what is non-trivial here? There are two problems. The first problem is that we are writing an intersection of things, okay?
07:10:880Paolo Guiotto: Well, do we have real intersection here? It means we are talking about taking F, sigma algae that contains S.
07:19:460Paolo Guiotto: I bet. He's happy.
07:23:140Paolo Guiotto: Because if there are no effort, we are defining nothing. Okay, you see the problem?
07:29:120Paolo Guiotto: If there are no sigma that contains X, what is sigma X that is intersection, or what is guys that do not exist? You may say it's nothing, it's… no, it's not defined, because there are no elements for which I can do that operation. So what I have to check here is that
07:48:460Paolo Guiotto: Since… So, let's say we, Check.
08:00:520Paolo Guiotto: That, huh?
08:03:680Paolo Guiotto: The intersection of all the… F, such that F is a sigma algebra that contains S mates… sense.
08:17:980Paolo Guiotto: That is… When this makes sense, there exists at least 1.
08:29:710Paolo Guiotto: F… sigma algebra, such that this F contains S. That's what we have to prove.
08:40:289Paolo Guiotto: Because once you have this, it means that there are F, so you can do always the intersection of sets. This is a set operation.
08:48:110Paolo Guiotto: Okay, now, the point is that,
08:52:210Paolo Guiotto: we say that there is at least one F of this type, which is part of X.
08:57:580Paolo Guiotto: F equals parts.
09:00:210Paolo Guiotto: of X is a sigma algebra, we checked above, and…
09:08:180Paolo Guiotto: S is contained into parts of S, of X, which is our F for this case, okay?
09:17:270Paolo Guiotto: So there exists at least one.
09:19:490Paolo Guiotto: Good. Now, this is… this explains, and I have nothing else to explain here.
09:25:340Paolo Guiotto: Okay, there is no other data.
09:28:610Paolo Guiotto: possible issue with that definition. Now, what we have to check is that it is a sigma algebra
09:35:600Paolo Guiotto: And, well, I forgot to say, it is a sigmatic, but that… contains S.
09:44:380Paolo Guiotto: And actually, it is the smallest one, because if we take any other sigma algebra that contains S, it must be contained into sigma S. Sorry, I have to add here, and…
09:56:960Paolo Guiotto: Let's say, moreover, Moreover… if, F tilde is a sigma algebra.
10:09:570Paolo Guiotto: that contains S, then F tilde contains also sigma S.
10:17:710Paolo Guiotto: Sorry, this horrible. Well, all this is trivial, it's not a particularly hard theorem to prove.
10:24:910Paolo Guiotto: Okay? There is no difficulty, it's just reading what is written.
10:29:940Paolo Guiotto: Okay? So, we checked that sigma S is well-defined, so this is the statement 1. So, this is, here, this part.
10:43:460Paolo Guiotto: here.
10:46:420Paolo Guiotto: Okay, this is the blue part. Now, let's check that it is a sigmada, which is the second statement down here.
10:54:700Paolo Guiotto: Which is what I wrote here.
10:57:340Paolo Guiotto: Okay?
10:59:450Paolo Guiotto: Well, actually, this is part of… Okay.
11:02:520Paolo Guiotto: Now, let's do the yellow part.
11:06:240Paolo Guiotto: Skip, what I don't know Okay.
11:09:330Paolo Guiotto: So, sigma S…
11:14:640Paolo Guiotto: is a sigma algebra. So, to check this, we have to go back to the definition once again. So, the three axioms.
11:23:690Paolo Guiotto: So, number one, MT and X belongs to sigmites. Why?
11:31:810Paolo Guiotto: Well, remind that sigma S, by definition, is the intersection of all the sigma algebra F,
11:38:340Paolo Guiotto: that contains the family S.
11:42:00Paolo Guiotto: So what I can say? Well, since F containing S, F sigma algebra.
11:50:980Paolo Guiotto: If I pick one of these sigma algebras I use to do the intersection, since it is a sigma algebra, I have that empty, and X belongs to F.
12:03:670Paolo Guiotto: And this happens for every F, because F is a sigma algebra, so X number 1 for F says that MTX belongs to all these Fs that I use to define sigmas.
12:17:270Paolo Guiotto: Since day belongs to all DF, they belongs to an intersection.
12:22:660Paolo Guiotto: Okay.
12:23:760Paolo Guiotto: So, from this it follows that empty and X belong to the intersection of all the F, F sigma algebra.
12:33:760Paolo Guiotto: such that F contains S, and that's exactly, by definition, sigma S.
12:40:770Paolo Guiotto: So, at the end, I have empty index belongs to sigma.
12:47:130Paolo Guiotto: Check number two.
12:50:210Paolo Guiotto: I have to prove that. If E belongs to sigma S, then, this is what we need to prove, E complementary belongs to sigma S.
13:04:100Paolo Guiotto: Now, the argument is similar. What do I know? I know that he belongs to sigma S. What is sigma S? It's that intersection that you see here.
13:15:110Paolo Guiotto: So if he belongs to the intersection, means that he belongs to each of these Fs.
13:20:260Paolo Guiotto: So this means that E belongs to F. For every F, Sigma. Algebra.
13:28:890Paolo Guiotto: that contains our original set, S.
13:34:160Paolo Guiotto: But F is a sigma algebra, right?
13:37:610Paolo Guiotto: F is a sigma algebra.
13:40:730Paolo Guiotto: So, since E belongs to F, also E complementary belongs to F.
13:46:890Paolo Guiotto: And this happens for every F, sigma, algebra, blah blah blah, right?
13:51:840Paolo Guiotto: But if E belongs to every sigma algebra that contains, S,
13:57:260Paolo Guiotto: Then, it belongs to the intersection of them.
14:00:660Paolo Guiotto: then E belongs to the intersection of all the F, F, sigma algebra.
14:09:680Paolo Guiotto: This is a kind of boring proofer.
14:13:600Paolo Guiotto: It's just we are doing what? We are working again on the definition, of this definition of what is a sigma article.
14:22:370Paolo Guiotto: Okay, it's abstract work, so it's a little bit harder, because you don't have a concrete example, numbers, things like this, but it makes us to manage this definition in order, we can learn without studying at the end.
14:40:760Paolo Guiotto: we will get so exhausted by doing this verification that we don't need to memorize the axioms, because we already practiced them. So this is sigmiles.
14:52:300Paolo Guiotto: And now, I'll do… so this proves our implication. Number 3, we should do the same argument for the union. Well, I leave to you. Let's start, then you finish, okay? So, in number 3, I have a sequence here of sets that
15:09:60Paolo Guiotto: is contained in,
15:11:790Paolo Guiotto: Sigma S, and the goal is to prove that their union is in sigma S as well.
15:22:230Paolo Guiotto: So, as you expect, we follow the same scheme. It seems this is a kind of straightforward argument.
15:29:730Paolo Guiotto: There is nothing to be invented here. So I have DN in the sigma algebra generated by S. What this means? It means that EN belongs to F, whatever is F, sigma algebra.
15:46:920Paolo Guiotto: containing S.
15:48:810Paolo Guiotto: And now, you continue.
15:51:140Paolo Guiotto: You easily arrive to the conclusion.
15:56:600Paolo Guiotto: You… Feesh.
16:01:640Paolo Guiotto: Okay, sold.
16:04:780Paolo Guiotto: This means that we know now that whatever is the family, we want to check if it is a sigma algebra, or if it is not, we want that those sets belong to some sigma algebra.
16:17:460Paolo Guiotto: We can always, of course, you can say, no, suppose that I return back to the… you do, I know, I have something else to put, and you finish also with the other part, let's give another call, sorry. I forgot that there is another part here.
16:33:20Paolo Guiotto: Let's see that this is the final part. You do this.
16:38:620Paolo Guiotto: Okay? You have to check it, that if it's… it's self-evident, because if F tilde is one of the sigma algebras, that contains S…
16:49:920Paolo Guiotto: What is F tilde? It's among those that I take in the third section, so it is clearly there, and therefore, sigma S will be smaller, because it is an intersection where there is also F tilde, so it's nothing to say. But also, so let's say that this was the proof of the yellow part.
17:07:970Paolo Guiotto: That, more or less finishes.
17:11:839Paolo Guiotto: here.
17:13:500Paolo Guiotto: You finish this… and also… also visa.
17:22:609Paolo Guiotto: books here for us, okay?
17:26:970Paolo Guiotto: Okay, now, one could say, okay, but let's go back to one of these examples. For example, let's take the example of the intervals, okay? We know that intervals, you know, they are not a family which is a sigma algebra.
17:44:00Paolo Guiotto: Okay? There is always a sigma algebra that contains intervals. Please.
17:57:240Paolo Guiotto: You can always say that there is a sigma algebra that contains whatever. Who is that sigma algebra?
18:06:30Paolo Guiotto: What's invites.
18:07:420Paolo Guiotto: The parts of, in this case, of R. All the subsets of R is a sigma algebra, contains intervals. Well, the problem is that the more you, you add into the sigma algebra, it means the more you have to be measured.
18:26:280Paolo Guiotto: And as you will see, this is actually a danger, because it can create some problems with the property of a measure. We will return to this when we will discuss
18:37:730Paolo Guiotto: probably tomorrow, the case of the LeBac measure, which is the reference measure for most of this course, for most of analysis, etc. So it's the important case.
18:49:220Paolo Guiotto: That measure, as you will see, starts by… with the idea of measuring everything. So, we can give a measure to every set of R2, R3, RM in general. Okay, nice, we can measure every set. But unfortunately, it turns out that this is impossible.
19:07:200Paolo Guiotto: So there is a theorem which is really hard and counterintuitive that shows that it is impossible to have a measure that has the features of the bag measure.
19:19:690Paolo Guiotto: We will see, which is an additive measure that fulfills that condition we anticipated above. The condition that the measure of the disjoint union is the sum of the measure. We will return on this. So, too many sets is actually a problem.
19:37:970Paolo Guiotto: So, the solution to the problem, I want that my family, my preferred family, like here, the family of intervals, B, in some sigma algebra, is to take the cheapest possible sigma algebra that contains that.
19:54:60Paolo Guiotto: So the reasonable sigma algebra to use is not part of X, which is the trivial sigma algebra that contains everything, so also intervals, but rather the sigma algebra generated by intervals.
20:08:600Paolo Guiotto: But this is a remarkable Sigmajiba, which is called the Boreal Sigmajib, but we will, you know, return a bit better later.
20:16:40Paolo Guiotto: So, it's an important concept, this one.
20:20:510Paolo Guiotto: Okay, let's say that we, for the moment, have introduced the concept, the definition of what is a sigma algebra, and this is the kind of family on which measures are defined. So we introduce now the
20:37:110Paolo Guiotto: Second fundamental definition for today, which is the definition of measure.
20:43:570Paolo Guiotto: Or better, abstract measure.
20:53:850Paolo Guiotto: So, we have a set X,
21:00:820Paolo Guiotto: And we have also a family F, which is a sigma algebra of sets of X. So F, B,
21:09:990Paolo Guiotto: Sigma. Algebra.
21:13:710Paolo Guiotto: Awful.
21:15:320Paolo Guiotto: sets… of X.
21:19:880Paolo Guiotto: a function.
21:23:790Paolo Guiotto: Because the measure at the end is a function, mu, defined on F.
21:29:00Paolo Guiotto: With values in 0 plus infinity.
21:32:420Paolo Guiotto: plus infinity included. We will need to say something about the algebra of plus infinity, but nothing special, we will tell later.
21:41:400Paolo Guiotto: In a moment, even if we should do before to give the definition. A function mu defined on this family F, with values in 0 plus infinity.
21:49:850Paolo Guiotto: is colder.
21:55:80Paolo Guiotto: measure.
22:00:270Paolo Guiotto: If… well, there are just the two axioms we have written above. So, number one is that the measure of nothing must be zero. This, as you understand, is not zero.
22:13:230Paolo Guiotto: particularly, say, crucial, but it's, it's a coherence requirement that we, that we put. And the second one is the really unique axiom for this definition.
22:28:150Paolo Guiotto: The surprising fact is that we don't need any other particular property. So, apparently, it's simple, but in fact, it's complicated. So, this property is called countable
22:44:170Paolo Guiotto: additivity.
22:50:30Paolo Guiotto: So it, it, it happens that, for every family, EAN,
22:57:260Paolo Guiotto: of subset of the sigma algebra F.
23:02:230Paolo Guiotto: Such that… They are disjoint, so EN, intersection, EM, equal nothing for N.
23:12:540Paolo Guiotto: different… from M… Then, Well, as we know, since F is a sigma algebra.
23:22:320Paolo Guiotto: We know that it is closed with respect to each countable union, so not only for disjoint unions, which is the case of this one. We are doing the union of the yen in a moment.
23:34:310Paolo Guiotto: But in particular, since the N are in the family F, the union of the N is in the family F, I don't have to write, because this comes from F is a sigma algebra.
23:44:450Paolo Guiotto: So, what happens is that the measure of the union of the yen
23:50:610Paolo Guiotto: Is the sum of the measures
23:54:590Paolo Guiotto: of the N. So I don't have to say the union is in the family F, because this is… this comes already with the fact that F is a sigma algebra.
24:07:280Paolo Guiotto: Okay, now, for convenience, to avoid every time to carry this specification that the sets must be disjoint, we will denote disjoint units with a symbol, which is a slight modification of the classical union notation.
24:30:260Paolo Guiotto: We will use the sort of squared union.
24:34:950Paolo Guiotto: This stands for the Classical Union.
24:38:530Paolo Guiotto: But provided the sets are disjoint.
24:42:900Paolo Guiotto: when…
24:45:950Paolo Guiotto: EN intersection EM is empty for N different from N. So it's the union, but it's just a notation that reminds of the fact that they are disjoint.
24:58:430Paolo Guiotto: Okay, so we will use the classical union rounded symbol for genetic unions.
25:03:860Paolo Guiotto: not necessarily disjoint, but if you want to use the disjoint union, we use this symbol to avoid writing the other part, and so we will write that mu of a disjoint union in this form.
25:18:890Paolo Guiotto: Is just that this sum.
25:21:470Paolo Guiotto: of the measure.
25:24:640Paolo Guiotto: Now, as I told you.
25:27:270Paolo Guiotto: There is a little… a couple of little remarks to do, because we have a sum there, that's a sum of numbers, and first of all, this is an agenda, an infinite sum, because the sector has been too many.
25:43:170Paolo Guiotto: Well, this is not a big problem, because you have learned in the first year class that there is disparation. We find sum in the sum of the CMS, okay? Here, we have to add a sum, specification, because this quantity can be even equal to classically. So what happens if you have a sum
26:02:730Paolo Guiotto: I mean, possibility. Passability is not a number. But here, you agree easily this, So that…
26:12:880Paolo Guiotto: The sum over N.
26:15:230Paolo Guiotto: of mu EN, ease… the… some.
26:23:120Paolo Guiotto: of a serious with the…
26:30:320Paolo Guiotto: formally, I should say non-negative, but I prefer to say positive, okay? Even if they are zero, we call them positive. With positive.
26:41:550Paolo Guiotto: firms.
26:45:140Paolo Guiotto: And by definition, The meaning of that infinite sum
26:53:530Paolo Guiotto: So let's write with all the required details, sum for N going from 0 to infinity of mu p.
27:03:190Paolo Guiotto: For the… by definition, this is what?
27:06:20Paolo Guiotto: Well, we don't know how to do the sum of infinitely many numbers, because the sum is a binary operation. We sum 2, then we add another one with some 3, then we add another one with some 4. We can do any finite sum, but we cannot do the infinite sum. So, how to solve this? Well, the idea is we take the limit.
27:24:620Paolo Guiotto: Sending the number of terms in the sum to plus infinity.
27:29:120Paolo Guiotto: And we do find its sum, so sum for N going from 0 to capital N of the mu AN.
27:37:700Paolo Guiotto: Now, so… we will use this problem to computing limits of things which are well-defined.
27:48:560Paolo Guiotto: The problem here is that what is some of this cell is plus infinity? We don't have the algebra for that. But, since all of them are positive, we may imagine that if we have a number plus plus infinity, the result is plus infinity. With the agreement, so…
28:06:490Paolo Guiotto: Off.
28:10:650Paolo Guiotto: So, wheat.
28:15:610Paolo Guiotto: agreement.
28:22:110Paolo Guiotto: that… If we have a positive number, so here A is greater or equal than zero a real number.
28:31:990Paolo Guiotto: And we add the plus infinity.
28:35:790Paolo Guiotto: So the result of this sum is plus infinity.
28:40:250Paolo Guiotto: Okay?
28:42:110Paolo Guiotto: And the clearly limit of a sequence equal to plus infinity will be plus infinity.
28:51:940Paolo Guiotto: issue of what if one of the terms is passivity. No problem.
28:57:150Paolo Guiotto: And another remark that's remind of what happens to Sirius, here, this is a series with the positive data, because these are measures. For us, they are positive numbers.
29:12:950Paolo Guiotto: Now, you know that the infinite sum can have troubles, because this field, as a limit, can do three things. It exists finite, it exists infinite, it does not exist.
29:26:190Paolo Guiotto: Oh, bye.
29:27:330Paolo Guiotto: 1 minus 1, that's the classical example, no?
29:30:610Paolo Guiotto: But the third term cannot be, because this series has positive terms. So either that limit is finite, either it is plus defined. So it is always defined, whatever are the numerics. So we are sure that whatever are the terms, that sum is always well defined.
29:50:180Paolo Guiotto: You don't have to care. Is it convergent? Is it divergent? B, if you want to know the value gas, we should take care. But, there is no question about convergence, divergence, and,
30:02:400Paolo Guiotto: Indeterminacy, which is the term that therapy, which is not present here.
30:07:470Paolo Guiotto: So, let's say also this. Moreover…
30:14:660Paolo Guiotto: Since, terms, new yen.
30:18:880Paolo Guiotto: Greater to equal 0 for every N.
30:22:320Paolo Guiotto: serious… Sum.
30:28:320Paolo Guiotto: of it, and… U-E-N… is always.
30:37:420Paolo Guiotto: convergent.
30:39:80Paolo Guiotto: or divergent.
30:41:620Paolo Guiotto: Okay, but in any case, it is always well-defined.
30:46:80Paolo Guiotto: That is not the problem. The limit does not exist, so we don't know what is the value of this sign.
30:51:220Paolo Guiotto: Okay, so, since we have just a few minutes, Let's start with an example.
31:01:240Paolo Guiotto: With a couple of simple examples, I would say.
31:05:610Paolo Guiotto: So, example one of measures, of course.
31:10:300Paolo Guiotto: Example.
31:15:990Paolo Guiotto: This… the both are easy, the second one is a bit less easy.
31:21:260Paolo Guiotto: This is the so-called, DRAC That is the manager.
31:30:590Paolo Guiotto: I don't want to tell you the story of theta wave, maybe we return later on this, but what is this measure? Well, take any set X,
31:42:350Paolo Guiotto: So this measure is a measure that can be defined wherever you like, any set.
31:49:580Paolo Guiotto: Take F, be any sigma algebra, Sigma algebra.
31:56:920Paolo Guiotto: And, define the measure in this form.
32:03:310Paolo Guiotto: So, pick a point, X.
32:06:600Paolo Guiotto: A pointer in a capital letter, fixte.
32:12:470Paolo Guiotto: And defined, this is usually denoted with this symbol, delta X, defined on F with values. Actually, you will see that the values are just 0 or 1.
32:23:600Paolo Guiotto: But we write 0 to plus infinity.
32:27:390Paolo Guiotto: And we, we define the delta X in this way. Take a set E in the family F,
32:35:480Paolo Guiotto: And you give value 1 if X belongs to E.
32:40:450Paolo Guiotto: and 0 if x is not in E.
32:44:580Paolo Guiotto: So this is also called an indicator function. Basically, it says the point is in the set or not?
32:51:110Paolo Guiotto: But let's look as function of the settee, and it is easy to check that this is a measure on the space. You see, it's a bit trivial measure, because the values assigned are only 0, 1.
33:05:210Paolo Guiotto: this mu, delta X, is, man, sure.
33:13:390Paolo Guiotto: Well, if I have to check that this is a measure, what I have to do? I have to check that, well, is this function defined, well-defined?
33:24:980Paolo Guiotto: is a function, you have to be careful. Does it make sense, this? Yes, no, there is no problem. You take a set, you suppose that you are always able to say if an element is in the set or not, so there is no question about the good position of these details.
33:41:320Paolo Guiotto: So, now what remains is to check the two axioms.
33:46:380Paolo Guiotto: Measure of empty 0, and the measure is countably additive.
33:52:740Paolo Guiotto: Okay, so, let's check…
33:58:500Paolo Guiotto: Neat.
33:59:740Paolo Guiotto: axioms.
34:03:630Paolo Guiotto: So number one, what is delta X of empty?
34:08:870Paolo Guiotto: Now, if I pick a set empty.
34:11:820Paolo Guiotto: what can I say is 1 or 0? Is 1 if the point is in the empty set, and that cannot be, because the empty set has nothing, or it is 0 if the point is not in the empty set, which is the case. So, this is 0.
34:27:489Paolo Guiotto: Because it's allowed by dystrophy.
34:30:350Paolo Guiotto: So, the first condition has been verified, quite trivially. The second one
34:36:199Paolo Guiotto: seems to be more complicated. I have to do the delta X of a disjoint union of subsets.
34:44:920Paolo Guiotto: And now, I have to see if this is…
34:48:760Paolo Guiotto: the sum of the delta X of the e-act.
34:55:670Paolo Guiotto: Okay.
34:56:840Paolo Guiotto: This is not immediate, not so complicated, but we have to think a set. Now, let's think the left-hand side. We know that this is a data observability, there's zero one, there is no other observability.
35:09:980Paolo Guiotto: Okay? So now, when is it zeros?
35:15:60Paolo Guiotto: This one is zero, if… Okay, okay, so delta X is equal to zero, Of the square union.
35:28:490Paolo Guiotto: is 0, if and only if X is not in the union, right?
35:34:530Paolo Guiotto: But this means that…
35:41:40Paolo Guiotto: Exactly, so X is not in EN for every N, otherwise it would be in the union.
35:50:580Paolo Guiotto: But then, this means that, what about the delta X of yen?
35:56:680Paolo Guiotto: It's zero for every amp.
35:58:990Paolo Guiotto: And this means that, well, at least this way is clear.
36:03:690Paolo Guiotto: If all delta XEN are zero, when I sum up, I get zero, okay? So the sum of the delta XEN
36:13:530Paolo Guiotto: is equal to zero. I think you agree on this.
36:16:930Paolo Guiotto: Is that also through the vice versa?
36:20:880Paolo Guiotto: Because think about, what if this summons here?
36:24:590Paolo Guiotto: Well, this is a special summoner. This is something that you have to keep in mind. You are summing positive numbers for the, and you get zero.
36:34:240Paolo Guiotto: This is possible only.
36:38:810Paolo Guiotto: All numbers are zero. Is one of them positive?
36:43:140Paolo Guiotto: No? This is not true, no? It's not true, there are… by some numbers, you get zero, the numbers must be zero, no? Plus 1 minus 25 to 0. But, since they are positive, they must be all equal to zero, so I can actually say that this is a different or equal.
37:02:50Paolo Guiotto: Okay, so delta is zero even if the sum is zero, so the same value, and you can easily see that the same holds for
37:11:620Paolo Guiotto: The value equal to 1.
37:14:640Paolo Guiotto: Because what… what happens to be…
37:17:780Paolo Guiotto: delta X of the union equal to 1. Of course, X must be in the union, and that's clear.
37:27:30Paolo Guiotto: But…
37:30:220Paolo Guiotto: Just one, because they are disjoint, so you cannot be in two. Otherwise, the two would have a common point. So there exists a unique N,
37:44:970Paolo Guiotto: a unique, So I don't know if you are familiar with this, but Xist… exists
37:55:840Paolo Guiotto: unique.
37:58:540Paolo Guiotto: And such that X belongs just to that EM, and none of the others.
38:05:300Paolo Guiotto: So it means that you have that delta X for that y n is 1.
38:12:210Paolo Guiotto: But delta X for all the other EN is 0 for every little n different from that capital n.
38:22:780Paolo Guiotto: Now, if you do this, and then you take the sum, you have one
38:28:300Paolo Guiotto: in one term of the sum, and zeros everywhere else. So the sum is 1.
38:33:730Paolo Guiotto: So the sum… of delta X.
38:37:580Paolo Guiotto: Yeah, no?
38:38:860Paolo Guiotto: Can be equal… is equal to 1.
38:42:750Paolo Guiotto: And also vice versa, because here, you also make numbers which are either 0 or 1s. So what can be that sum?
38:50:870Paolo Guiotto: tenant.
38:54:450Paolo Guiotto: That could be one, huh?
38:56:400Paolo Guiotto: Means that only one of them must be 1, and though the others are 0.
39:02:850Paolo Guiotto: I can actually say that that's an if and only if.
39:06:870Paolo Guiotto: Okay, so we checked the accountability.
39:10:840Paolo Guiotto: Let's stop here, maybe tomorrow we see some other examples.
39:17:00Paolo Guiotto: Okay.
39:24:680Paolo Guiotto: Well, actually, I will write down here some exercises you could start thinking about.
39:30:730Paolo Guiotto: do exercise. They are at the end of the chapter, 1 for 1.
39:38:100Paolo Guiotto: The number 2… The number 3… the number 4…
39:49:440Paolo Guiotto: But I would say that this is sufficient. Okay.