AI Assistant
Transcript
00:01:780Paolo Guiotto: Okay… But not the previous life.
00:20:860Paolo Guiotto: There is another solution, but I do not remember, last year was the same problem.
00:33:130Paolo Guiotto: Okay.
00:35:480Paolo Guiotto: So let's try to put the argument in this way. So, from this.
00:43:610Paolo Guiotto: We can say that So… Either…
00:52:510Paolo Guiotto: all D, E, N are comfortable, so EN is… Countable.
01:01:440Paolo Guiotto: For every year, no?
01:04:470Paolo Guiotto: Let's say the alter… let's write down the alternative first. So, either this happens, or…
01:10:390Paolo Guiotto: What is the contrary of this?
01:13:830Paolo Guiotto: Well, the opposite is not all the Naran are…
01:18:920Paolo Guiotto: Okay, it's clear, no? Because the alternative is not all countable, all the n countable, or all the n complementary countable, but…
01:36:360Paolo Guiotto: all the N countable, or one of them is not countable, but then there is the complementary which is countable.
01:44:350Paolo Guiotto: So, at least one… so, there exists, let's say, an N, let's call capital N, such that EN complementary is countable.
01:58:420Paolo Guiotto: Okay.
02:00:380Paolo Guiotto: Now, if we are in the first case, we can say that then the union of the yen is countable.
02:09:110Paolo Guiotto: And this is,
02:11:530Paolo Guiotto: The conclusion, one of these two. Now, what about the second case? Of course, we cannot ensure… the union of the N won't be countable, maybe, but let's see the alternative. The alternative is this set. What can be said about this?
02:28:390Paolo Guiotto: Whoa.
02:30:00Paolo Guiotto: Got it.
02:31:380Paolo Guiotto: the intersection is contained in the area, so… Yes, exactly. So the full intersection is contained in each of the sets, particularly to this one. So, we can say that the intersection
02:50:700Paolo Guiotto: over n, of the n complementary, is contained in that n complementary, and since this guy is in correspondence with natural, the other one which is smaller will be in correspondence with natural, okay? Because it's smaller. So, this implies that
03:10:40Paolo Guiotto: This is countable.
03:12:290Paolo Guiotto: This implies that the intersection of the yen Complementary is comfortable.
03:18:910Paolo Guiotto: Okay, so this, ends the proof.
03:22:210Paolo Guiotto: of decent.
03:24:180Paolo Guiotto: problem, and therefore, we, we have the conclusion. Our family, defined in this way, is a Sigmatic, okay?
03:33:990Paolo Guiotto: Now, I repeat, this is not particularly
03:38:630Paolo Guiotto: Important example is just a pedagogical example to use to learn to use the definition of sigma algebra.
03:47:520Paolo Guiotto: Even because, as you will see, I already anticipated this, it's quite hard to build directly Sigma algebras, okay? In fact, let's see some examples, some natural examples of natural sets.
04:03:880Paolo Guiotto: which, are not familiar sets, which are not sigma algebras. So let me take this example.
04:13:700Paolo Guiotto: So, examples… of… Families.
04:27:230Paolo Guiotto: notes.
04:29:410Paolo Guiotto: Sigma algebra. Of course, I mean.
04:32:500Paolo Guiotto: you can take any family which is not algebra, it's good, but these examples are interesting because we take, let's say, natural exam, the families of sets. For example, here we are, in R,
04:48:860Paolo Guiotto: So X is the set of real numbers. An actual family of sets that we used since the first year are intervals. So, let's say the family of sets I
05:03:460Paolo Guiotto: contained in R with our, interval.
05:08:00Paolo Guiotto: I didn't forgot.
05:09:850Paolo Guiotto: I do not write a formula, but I can be, I don't know, the interval AB open, or ABF open on one side.
05:21:740Paolo Guiotto: On the other side, Closed.
05:27:120Paolo Guiotto: I don't want to list all these types, but when we say interval, we mean it could be an unbounded interval, so an offline, like 8 plus infinity, closet at left, open at left.
05:41:970Paolo Guiotto: I don't know what is the notation you are used to, to denote open sets. Sometimes, for example, another very popular notation is this one.
05:52:740Paolo Guiotto: To use rounded parentheses.
05:55:140Paolo Guiotto: Okay? I don't like this one because it confuses with points in the plane, so maybe…
06:01:360Paolo Guiotto: I prefer this with the square brackets, but whoever you can use the notation you prefer. Then we can have, I don't know, the intervals, something like minus infinity to B, open, minus infinity to B,
06:17:730Paolo Guiotto: Closed, and also, the full interval minus infinity plus infinity, which is just… Are the real life.
06:29:320Paolo Guiotto: So, intervals means all this family of things for all possible A, B, etc, okay?
06:35:520Paolo Guiotto: Now, this family is not a sigma algebra.
06:39:480Paolo Guiotto: Well… Let's see why.
06:42:590Paolo Guiotto: F… is not.
06:47:610Paolo Guiotto: Okay.
06:49:340Paolo Guiotto: Sigma algebra.
06:53:240Paolo Guiotto: So… Before we write down an argument.
06:58:130Paolo Guiotto: What… which of the three conditions
07:02:550Paolo Guiotto: that characterize a sigma algebra, no? These three.
07:07:240Paolo Guiotto: Which one, or which ones… Are not true in this case.
07:12:280Paolo Guiotto: What do you think?
07:13:640Paolo Guiotto: The first, the second, or the third.
07:17:410Paolo Guiotto: It does not contain empty or the full set, it is not closed for the complementary, or it is not closed for unions.
07:26:740Paolo Guiotto: Where are problems here?
07:31:90Paolo Guiotto: What do you think?
07:35:730Paolo Guiotto: Okay.
07:37:100Paolo Guiotto: Let's see. Does it contain the empty set?
07:45:820Paolo Guiotto: Yeah, empty set is the interval.
07:49:180Paolo Guiotto: Yeah, but write as an interval. For example, I have an interval from 0 to 0 open. There is nothing between 0 and 0, okay?
07:57:710Paolo Guiotto: So, and the full space, X, is the realignment, which is the interval minus infinity to plus infinity. So, the first condition is verified.
08:08:280Paolo Guiotto: The number 2 is verified, this one, is it true?
08:13:960Paolo Guiotto: The complementary of an interval, is it an interval?
08:19:780Paolo Guiotto: It depends, okay? So, when do you have an interval?
08:24:200Paolo Guiotto: By doing the complementary of an interval. For which kinds?
08:29:510Paolo Guiotto: Yeah. Exactly. When it gives enough flat, no? If you do the complementary of this one.
08:41:240Paolo Guiotto: minus infinity. To B, you get something like this one, B to plus infinity.
08:46:500Paolo Guiotto: So it is true that the complementary of this is this. The complementary of this, means something like that, and so on. But, if you do the complementary of one of those.
08:58:300Paolo Guiotto: You get 2 interval.
09:00:450Paolo Guiotto: Okay? Because if you do the complementary of, I don't know, AB, think about AB is this.
09:07:940Paolo Guiotto: Including, the endpoints is this one.
09:12:900Paolo Guiotto: So this is the interval AB.
09:16:40Paolo Guiotto: What happens when you do the complementary? It's the black line, you know? So you get…
09:21:750Paolo Guiotto: an offline here that excludes A, and here from B to plus infinity.
09:28:230Paolo Guiotto: So… The green set is the complementary.
09:32:580Paolo Guiotto: of AB, which is minus infinity.
09:36:450Paolo Guiotto: way.
09:37:790Paolo Guiotto: union from B to plus infinity. So we can stop here. We don't need to check that it is not close with respect to countable unions. Even that property is not verified, because even just if you take two intervals, you do the union, you do not necessarily get an interval. If they are disjoint, they remain disjoint, okay?
09:58:350Paolo Guiotto: So, this is sufficient.
10:01:300Paolo Guiotto: Okay, we may think, then, that, you see that the complementary of an interval is not an interval, but it is a union of intervals. So what if we extend this example in this way?
10:16:00Paolo Guiotto: take still X equals R, but now modify F.
10:22:570Paolo Guiotto: Now, why we are doing this? Because, normally, these families remind that the goal is, these are the domains of the measure. So, the family of sets to which we can assign a measure, if this measure has geometrical meaning, area, volume, whatever.
10:39:670Paolo Guiotto: Or a probability.
10:41:780Paolo Guiotto: Okay? Could be the measure will be interpreted as a probability, the value of the probability that that set is verified, okay?
10:51:440Paolo Guiotto: So normally, what happens when you build a model is that you start defining what are the sets that you want to measure.
11:00:690Paolo Guiotto: Because you want to assign a measure to this class of sets that you like, or you need for certain hypothesis. So normally, you start from a class of good sets that you would like to have in the family.
11:13:780Paolo Guiotto: Normally, if you take a class of sets.
11:18:110Paolo Guiotto: this hardly would be a sigma algebra, because as here, I like intervals, but I discovered that they are not close respectively complementary. So, okay, let's extend the class in such a way that this is verified. What if I take the family that is made by unions of intervals?
11:37:720Paolo Guiotto: Okay? So let's say, well, to stay on,
11:43:910Paolo Guiotto: on which unions. We stay here on countable unions. In this way, if I take a countable union of intervals, automatically, when I do a countable union of intervals, I am inside.
11:55:420Paolo Guiotto: the family, okay? So take,
11:59:230Paolo Guiotto: these kind of sets, which are INR, so with the N natural, intervals.
12:17:680Paolo Guiotto: Now, is this a sim algebra? Now, this is a 3-star problem.
12:23:940Paolo Guiotto: which is based on some fine arguments related to, uncountability of the real line, okay? So, I don't want to… there is the solution there, but maybe we could… we don't look at the solution, you could try to think about this problem.
12:42:460Paolo Guiotto: So, he's, F.
12:45:470Paolo Guiotto: a sigma algebra. Of course, here we are in the class of example, it is not a sigma algebra, so you already know that it is not a sigma algebra.
12:56:00Paolo Guiotto: But there is no difference. How should the… what is the difference in checking if it is a sigma algebra or checking that it is not? Well, you start normally, you don't know if it is a sigma algebra, so you try to prove that it is a sigma algebra.
13:08:690Paolo Guiotto: At a certain moment, maybe you arrive to a difficulty, you are not able to prove a certain step, and then maybe it could suggest to you that, okay, let's try for a counterexample, okay, here.
13:21:560Paolo Guiotto: Okay, now, to answer, it's a bit difficult, so I put 3 star.
13:26:280Paolo Guiotto: Try to do… we can… we will see tomorrow, so try to think about this problem.
13:33:740Paolo Guiotto: It's a little bit, complicated.
13:38:690Paolo Guiotto: Okay, so…
13:41:440Paolo Guiotto: Another popular example, this is an important example, in fact, we take, in general, in our M, we take as family F,
13:53:200Paolo Guiotto: An important class of sets, important sets are those sets that you know from analysis. Open sets, closed sets, and so on, okay? Let's take the class of open sets, so the sets O contained in RM, such that O
14:11:700Paolo Guiotto: is open.
14:15:30Paolo Guiotto: Open means that each point is contained into the set with a ball, okay?
14:23:580Paolo Guiotto: So, roughly speaking, there is no boundary for this set.
14:27:870Paolo Guiotto: Now, remind that for convenience, empty is considered an open set, by definition, because you cannot check anything. There are no points, you cannot check if the point is inside with the ball. So, what is the answer? Well, for convenience, we assume that empty is open.
14:46:180Paolo Guiotto: So, it is easy to see here that the empty and the space are both open, so we are inside them.
14:56:360Paolo Guiotto: So, here… empty and, X.
15:04:30Paolo Guiotto: are in the family F.
15:06:720Paolo Guiotto: But… if, an open set belongs to the family F, so if O is open.
15:17:200Paolo Guiotto: Well, in general, or complementary, is not open.
15:22:740Paolo Guiotto: Well, actually, the unique set, which is both him and it's complementary, they are both open, can be empty under full space.
15:34:250Paolo Guiotto: This can be proved. Okay, so there is no other set different from these two guys, that it is, at the same time, open with its complementary.
15:44:20Paolo Guiotto: Okay? So, all complementary is open.
15:51:40Paolo Guiotto: Well, this is non-trivial,
15:53:520Paolo Guiotto: Trust me. But here, we didn't have to think that this is impossible, just we need an example. Well, actually, if and only if O equal empty or X. But in general, you can realize that this is false.
16:09:620Paolo Guiotto: Because if you, for example, just to keep the example in R1, X equals R1,
16:18:120Paolo Guiotto: take, as, O, the open set and open interval, A, B,
16:23:730Paolo Guiotto: like that, the complementary of this set is, as we know, the 2F line, minus infinity 2A,
16:31:350Paolo Guiotto: included, plus the offline from B included to plus infinity, and this site is not open.
16:38:950Paolo Guiotto: this, is… not.
16:43:150Paolo Guiotto: Open.
16:45:520Paolo Guiotto: And because it is not true that each point belongs to the set with a little ball that, in this case, is an interval, okay? The end points A and B
16:58:350Paolo Guiotto: They cannot stay with a little ball, because if you center a ball there, as you can see, you go outside of the set, so A and B are not contained with the ball.
17:11:470Paolo Guiotto: So this family, which is an important family for applications, is not, is not a single algebra.
17:20:210Paolo Guiotto: You may notice that, well, A, all complementary, if all is open, is what we call a closed set. So, what if I take the family made of open and closed set, which is
17:32:360Paolo Guiotto: Closed sets are important as the open sets. In fact, they are each the complementary of the other, so they are basically
17:40:870Paolo Guiotto: strictly related. Well, it happens that, of course, you don't get, again, a family which is a sigma algebra, because in general, it is true that if you take an open set, you do the complementary, you get closed.
17:54:890Paolo Guiotto: Really do a union of two sets, one open, other closed. Normally, you get a set which is neither open.
18:03:140Paolo Guiotto: No clothes, okay? So, also, let's say…
18:07:800Paolo Guiotto: Maybe you can think about, also.
18:11:720Paolo Guiotto: F equal the family of sets E, such that E is open.
18:21:500Paolo Guiotto: or it is.
18:24:500Paolo Guiotto: Close the… is… not.
18:30:910Paolo Guiotto: at Sigma.
18:35:900Paolo Guiotto: Okay.
18:37:750Paolo Guiotto: So, check out the details.
18:41:300Paolo Guiotto: Oh, it's 11.24. Do you want to take a short break?
18:48:410Paolo Guiotto: Okay, so if you want to say, let's take 5 minutes, is it okay?