AI Assistant
Transcript
00:19:510Paolo Guiotto: Thank you.
00:29:170Paolo Guiotto: Beautiful thing, Bristol.
00:51:720Paolo Guiotto: Okay, good morning.
00:55:10Paolo Guiotto: Okay, today we really start, huh?
00:58:740Paolo Guiotto: And, our first, topic, is, abstract…
01:09:970Paolo Guiotto: measures.
01:17:910Paolo Guiotto: So, what is an abstract measure?
01:31:740Paolo Guiotto: Well, the idea is that it is an extension of the concept of area volume.
01:37:810Paolo Guiotto: For, geometical figure.
01:42:590Paolo Guiotto: So, it is… an extension…
01:52:750Paolo Guiotto: area.
01:54:740Paolo Guiotto: It's voluminous.
01:58:30Paolo Guiotto: So…
01:59:190Paolo Guiotto: Now we are going to introduce formally the definition, but as you will see, before we can define what is really a measure, we need to define the concept. So we have a set X,
02:11:930Paolo Guiotto: genetic setter.
02:17:00Paolo Guiotto: for geometic applications, it will be R2, are 3… But in general, RB… what I'll say, RM…
02:28:510Paolo Guiotto: But it can be whatever, actually.
02:31:800Paolo Guiotto: Then, a measure is a way to assign a number to a set, to say, to measure the size of the set, no? So we have to imagine that we have a quantity, mu of E, which represents the measure
02:50:730Paolo Guiotto: of E, where E is a certain subset of the set capital X. So, as you can see, mu of V is a number.
03:02:250Paolo Guiotto: This number, if we… there are also, the so-called silent measures, even complex valued measures, but here we will focus just on
03:12:810Paolo Guiotto: positive value measures, and it can take also the value equal to plus infinity. If I have to say. What is the area of the full plane? It's plus infinity. So we consider also the possibility that it can be equal to plus infinity.
03:28:150Paolo Guiotto: So you see, that mu is something that does this. It takes a set E, and it yields a number, mu of e, that belongs to 0 plus infinity
03:44:990Paolo Guiotto: and points included. So, in other words, mu…
03:49:270Paolo Guiotto: Anytime you have a rule that takes something that gives you something else, you have a function. So mu is a function.
03:56:250Paolo Guiotto: is a function.
04:01:150Paolo Guiotto: of,
04:03:520Paolo Guiotto: It's not a numerical function in the sense that it's not a function of a number as a variable. The variable is a set.
04:10:950Paolo Guiotto: Okay, of a set.
04:14:340Paolo Guiotto: offset.
04:19:730Paolo Guiotto: So we have two major that has every… for every function, there would be a domain, let's, denote this domain with,
04:27:660Paolo Guiotto: script F. Well, this, is the domain for the, for the mention U,
04:34:00Paolo Guiotto: So, it is what? It is an asset made of these objects, which are subset of X. So, this thing, the domain where this mu is defined, is a family of subsets.
04:50:620Paolo Guiotto: We say this, saying that this is a subset of the set of all the subsets of X, so it's what is called the parts of X.
05:01:60Paolo Guiotto: Let these ease up.
05:04:40Paolo Guiotto: bots.
05:06:580Paolo Guiotto: of XM, so set…
05:11:20Paolo Guiotto: Well, it's set of all subsets is functionalized, so let's say collection, but, you know, that collection, class, etc, they're all synonymous, but
05:20:340Paolo Guiotto: It's better, so, class, huh?
05:25:360Paolo Guiotto: aw.
05:27:150Paolo Guiotto: all… subsets.
05:34:330Paolo Guiotto: of tropical elements.
05:36:870Paolo Guiotto: So, the domain of this function mu will be a subclass of subsets of X.
05:46:70Paolo Guiotto: And, so this is the domain, and this mu yields, positive, possibly class infinity, values.
05:55:160Paolo Guiotto: That should fulfill, well, actually, at the end, that would be just, the, the, the…
06:02:380Paolo Guiotto: The needed properties are only two properties.
06:06:960Paolo Guiotto: Such that…
06:09:930Paolo Guiotto: Now, since among all these subsets of assets, there is the empty set, the set with nothing.
06:16:600Paolo Guiotto: But you know that this is a sort of mathematical agreement to make a certain number of things meaningful. You do the intersection of this joint set, what is this?
06:28:780Paolo Guiotto: Well, we call that the empty set. So, it's not the fundamental key, but what you expect be the measure of nothing, so…
06:36:650Paolo Guiotto: let's say the area of nothing will be zero. Okay, so the first condition is mu of MTB equals 0. This is actually, say.
06:50:590Paolo Guiotto: sort of convenience we do. But the most important condition is the second one.
06:57:60Paolo Guiotto: Well, the second one is the, in fact, the true condition that makes a measure a measure.
07:04:10Paolo Guiotto: And it says that when we have a disjoint union of parts, so if we can take a set, we can decompose into a disjoint union of parts. The measure of the total is the sum of the measures.
07:17:880Paolo Guiotto: Okay, so we could say, in a figure, if you have a set, say, E, that can be divided into disjoint parts, say, E0, E1, E2,
07:30:520Paolo Guiotto: and so on, E3. How many parts? Well, we could divide in a finite number of parts.
07:37:770Paolo Guiotto: And this could be a possible definition. The measure of a disjoint union, finite union is the sum, the finite sum of quantities.
07:49:610Paolo Guiotto: We can do a little bit more. If we think to geometry, it's natural to allow infinite unions, because, for example, imagine you want to fill a circle with rectangles. You cannot fill with a finite number of rectangles. You need an infinite number of rectangles.
08:06:240Paolo Guiotto: And, how many infinite? Well, there are many kind of infinities, okay? But, let's say the easiest one is the comfortable. Also, because here we have to say something like measure of the union equals sum of the measures, and that sum
08:22:220Paolo Guiotto: starts to be a problem if it is a sum different from a discrete sum. So that's why we have this condition, a measure of, I will say for the moment, this union of yen, but this union must be disjoint.
08:41:940Paolo Guiotto: So this means that if you take EM and you take EM with different indexes, the intersection is empty.
08:52:380Paolo Guiotto: So, it is natural that this be equal to the sum, the infinite sum, in this case, of E measured of E.
09:02:140Paolo Guiotto: SATS 8N.
09:03:920Paolo Guiotto: Now, that infinite sum is a series, is an infinite sum of numbers. They are positive numbers. Well, there could be some trouble if one of them is possibility, but it's very amazing if one of them or more of them, are people in classically, that sum would be plastically, because we are summing positive things.
09:25:790Paolo Guiotto: Otherwise, it's a series, and as a series can have a finite value, a series of positive terms, can have either a finite value or a positive final. So it's always either convergent or divergent. That is not the case when using the 100% of the cell.
09:44:450Paolo Guiotto: So, it makes sense.
09:46:320Paolo Guiotto: Okay, so this is the basic idea. Now, to…
09:51:230Paolo Guiotto: to write precisely this, as you can see, we need that. First, the empty set, on the antiset, we can compute mu, so the empty set must be in the domain of mu, so must be in the family F, where mu is the founder, so this is the first prediction.
10:10:300Paolo Guiotto: A second condition is this one. We need that this guy makes sense, and also this bond makes sense. So this means that the N must be in the class F, but also the union must be in the class F.
10:24:150Paolo Guiotto: Now, here it is a point that says, well, about the class F, we may assume, which is reasonable, that if I have sex in the class F, the union is automatically in the class.
10:37:830Paolo Guiotto: So the classics somehow close respect, and a union are set.
10:42:890Paolo Guiotto: accountable unit.
10:44:50Paolo Guiotto: Okay? So this means that the family F cannot be just a family of subset, it must respect these two conditions, and actually it turns out that there is a third condition that must be added here. So now, we focus on the family F before we can formalize the measure nu.
11:03:630Paolo Guiotto: And we introduced this, first important definition, let me take another glimpse at this.
11:11:410Paolo Guiotto: So, definition… And this is the definition of what is called a sigma algebra.
11:21:740Paolo Guiotto: Offsets.
11:27:900Paolo Guiotto: So letter X, beat.
11:34:170Paolo Guiotto: Sette.
11:36:170Paolo Guiotto: Genetic sector.
11:38:870Paolo Guiotto: a family… Of a class.
11:45:50Paolo Guiotto: F.
11:46:300Paolo Guiotto: contained into parts of X, so a family of subsets of X.
11:53:590Paolo Guiotto: is… Cold.
12:00:630Paolo Guiotto: Sigma algebra.
12:04:90Paolo Guiotto: If the following conditions are verified. If…
12:08:150Paolo Guiotto: the following condition of a file. Number one.
12:13:380Paolo Guiotto: Empty must be in the family.
12:18:400Paolo Guiotto: And as you will see automatically from… there is a little redundancy here from another condition. If empty is in the family, also the full space is in the family. But however, that's right, normally they are written in the first condition.
12:33:100Paolo Guiotto: they must belong… must be in the family. So, empty and X are there.
12:39:770Paolo Guiotto: Secunda.
12:41:730Paolo Guiotto: Well, about the… there is the condition on the… well, let's say that. Second is the condition on the complementary. If you have a member of the family.
12:52:890Paolo Guiotto: then also it's complementary must be in the family. Remind that this is going to be the domain of the measure.
13:01:230Paolo Guiotto: So, the sets to which we will assign a measure. If it is difficult, maybe at the beginning, think to area or volume, okay? But this will be also probability, because the probability measure is a measure in this sense.
13:18:170Paolo Guiotto: Okay, so sets to which we will assign a probability. Now, the idea is that why there is this condition, too. If you can assign a measure to, let's say, that imagine that you have the full axis, this one, and you can give a measure to this part, then you should be able to give a measure to the other part.
13:37:830Paolo Guiotto: Okay, that's the idea, it's quite natural, sir.
13:41:630Paolo Guiotto: Okay, but this is not true. As you will see, if you take a family of subsets, it's not automatically true that this is the fact. And number 3 is the condition on the union.
13:53:870Paolo Guiotto: so…
13:56:310Paolo Guiotto: we don't… we don't need to restrict this to disjoint unions, so we do a little bit weaker. And we say that if we have a family, Ian.
14:08:530Paolo Guiotto: where N can be natural, in F… They are enough, huh?
14:16:250Paolo Guiotto: So, since they… these sets will be called, in the language of measure, measurable sets, so let's think… let's start using this language. If we have measurable sets, and we put together, we do the, you know, we get an element still of the family, so a measurable set. Then.
14:34:510Paolo Guiotto: Their union, of the N belongs to F.
14:40:100Paolo Guiotto: So, in other words, a sigma algebra is a family of subsets which contains empty and the full space, and we say it is closed with respect to the complementary.
14:53:290Paolo Guiotto: Closed means it's not the… the topological closed, for those who are reminded what is a closed set, no? That's not the same sense, but it is saying, if we do the complementation, we remain in the class. And it is closed with respect to the union. If we do a countable union of objects, we still, which are in the class, we still remain in the class.
15:17:280Paolo Guiotto: Okay? So, if you want, later we will use the language measurable sets, we will call these measurable sets when we have a measure.
15:25:310Paolo Guiotto: And, this means empty at the full space are measurable. If a set is measurable, it's complementary, can be measured, it's measurable. And if we have sets which are measurable, we do the union, also the union can have a measure, okay?
15:40:80Paolo Guiotto: Not necessarily disjoint, I've not written here this.
15:44:240Paolo Guiotto: Okay, so this is the definition. Before we give also the definition of measure, the best thing to do is to do examples, from which you will understand, or you should understand immediately, something. So let's do examples.
15:59:320Paolo Guiotto: My daughter has changed all the colors, yeah. Okay. Example.
16:05:750Paolo Guiotto: Oh, I have not yet published the notes I will do today, and in the notes, you will see there are stars here and there in exercises, examples, and so on.
16:17:710Paolo Guiotto: Now, one star means is a basic thing, so it's easy, you have to understand, okay? Two star means it's normal. It's a sort of level of complexity, no? It's normal, but you can,
16:33:950Paolo Guiotto: You are expected to understand that.
16:37:100Paolo Guiotto: Even if it is more complex. And then there are something like 2 star plus, or 3 star, 3-star plus. Well, when there is the plus, means that it's a bit more complicated, maybe there is some technical complexity. Normally, 3-star refers to theoretical properties.
16:55:480Paolo Guiotto: Now, abstract things are more difficult to be understood than applied things, okay? So now, this is a one-star example.
17:03:680Paolo Guiotto: So we have, these, three examples. The number one is, take, so here is X is any sector.
17:15:430Paolo Guiotto: So take this family F, and this is called the trivial example, the trivial Trivial sigma algebra.
17:30:620Paolo Guiotto: It's the trivia because, as you see, the definition
17:34:690Paolo Guiotto: Every sigma algebra contains empty index. There are always
17:39:550Paolo Guiotto: Both, okay? So this is the simplest possible thing algebra that you can do that contains empty index. It's exactly made of empty index. And you can see, we don't need to write, we can do just by reading the definition. You can see that this verifies the definition.
17:57:790Paolo Guiotto: all contains empty. And the answer.
18:01:160Paolo Guiotto: If a cell is in the family, it's complementary is in the family.
18:06:310Paolo Guiotto: empty is in the family, the complementary is X, sorry, it's in the family. X is in the family, the complementary is empty. It's in the family. So, the second condition is qualified, and the third one, we cannot use such big unions here, because either all the yen are empty.
18:25:800Paolo Guiotto: No? So you're doing unions of empty set, you've got the empty set, which is in the family.
18:33:430Paolo Guiotto: Or, at least one of the yen is X. We need all the yen are X, but just one is sufficient, because when you do the union, and if one of them is the full space, you get the full space, okay? So, no matter how the yen are taken.
18:50:340Paolo Guiotto: Okay, they would be either all equal to empty, or at least one of them is non-empty, therefore there is one exit inside this unit. The union, the first case is empty, the second is empty. So, the three conditions are verified.
19:05:520Paolo Guiotto: Another trivial, but less trivial than sigma algebra, is…
19:13:940Paolo Guiotto: the biggest possible sum of algebra, which is exactly part of S, of X.
19:22:80Paolo Guiotto: Also, in this case, the check that this is a sigma algebra is trivial, because if you look at the definition.
19:30:130Paolo Guiotto: In parts of X, there are all these subsets of X, including empty and X itself.
19:37:230Paolo Guiotto: So, the first condition is verified.
19:40:670Paolo Guiotto: It is clear that if you take a set in this family, its complementary is still a subset of X, because if you take a subset of X, you do the complementary, you still have a subset of X.
19:53:300Paolo Guiotto: Okay? So, you are… in the family, end up for the second condition is very far. And third, whatever R, D, E, N, they are subset of X. You do the union of subset of X, you get a subset of X.
20:08:550Paolo Guiotto: So, extremely young.
20:10:900Paolo Guiotto: Let's see something less trivial.
20:14:440Paolo Guiotto: Now, the point that you will… you should understand here is that it is not easy.
20:21:500Paolo Guiotto: See, it seems an easy concept, this one, no? Family are subset with these three relatively simple conditions.
20:29:340Paolo Guiotto: But in practice, if you want to explicitly define a sigma album, it is really non-trivial, apart for trivial cases. This is basically the idea.
20:41:730Paolo Guiotto: And so this is not a good point to start, because we're starting with the concept, which is the bailment of what we are going to say, and… but to construct explicit example, it seems to be not easy, okay? However, as you will see, this is really not a problem.
21:00:930Paolo Guiotto: This third example is that take a subset A, which is a proper subset of X, and non-empty.
21:10:740Paolo Guiotto: Of course, there should be an example like that, so we assume that our X is not… is made at least of two elements to have this, no?
21:23:660Paolo Guiotto: So if you have elements A and B, you take the set made only of A. Now, the family in this case is made in this way. There is empty index, they are always there.
21:36:720Paolo Guiotto: There is A. Now, if there is A, there must be also its complementary, right? So, if I want to have a sigma algebra, there should be also A complementary.
21:46:230Paolo Guiotto: Now, I said that this is a sigma algebra.
21:50:250Paolo Guiotto: And in fact, you can see that this is true, because first, it contains empty index, as you can see.
21:59:120Paolo Guiotto: Second, if a set is in the family, also its complementary is in the family, that's true. That's the same algorithm. Before, if we take this one, its complementary is this one, so it's in the family. If we take this one, the complementary is this.
22:16:840Paolo Guiotto: So it is still in the family. And, so this verifies the condition 2. It is closed with respect to the complementation. And number three, if you do unions here.
22:28:420Paolo Guiotto: So, imagine that today, sets that can be NPX pay or pay incremental.
22:35:500Paolo Guiotto: We can easily see that work to begin union, because, they can be all empty. They can be exact.
22:42:950Paolo Guiotto: At least one of them is A, and there is only A, the union is A. There is also A complementary, there is A union A complementary that makes X, so the union is X.
22:55:520Paolo Guiotto: There is only a complementary entity sets, the union is a complementary.
23:00:820Paolo Guiotto: There is also X, the unit is definitely equal to X. So you see that whatever is the parity of sets big there, you cannot do so many units, you get one of these four sets. So this is an example of an intermediate sigma algebra.
23:20:370Paolo Guiotto: Well, let's see… this is a typical example that we find in old textbooks of major theory.
23:28:210Paolo Guiotto: So it's a little bit more complicated, so let's put the two-star example.
23:33:850Paolo Guiotto: Because we have to think about it a bit. Okay, it says, X, Well, any…
23:45:360Paolo Guiotto: set… yes, the example makes sense, but it's non-trivial when the set X is infinite, but you can… it works in any case, any set.
24:01:800Paolo Guiotto: This example, has no, practical purpose. It's just pedagogical to make practice on the definition, okay?
24:11:300Paolo Guiotto: And the family F is defined in this way. We take the sets E, which are contained in X, such that
24:19:270Paolo Guiotto: Either, well, either. At least…
24:27:90Paolo Guiotto: One… Oh, E.
24:31:100Paolo Guiotto: or e-complementary, is… Countable.
24:40:280Paolo Guiotto: You know what countable means?
24:43:150Paolo Guiotto: So, roughly, comfortable means that it is, like.
24:48:840Paolo Guiotto: the set of naturals. It can be… it can be put in correspondence one-to-one with naturals.
24:56:480Paolo Guiotto: Okay, naturals, of course, integers are comfortable. Also, rationals, this is at least a bit less intuitive, because rationals looks to be much more than naturals, but in fact.
25:09:90Paolo Guiotto: they are countable. So it's in correspondence with the hand.
25:14:290Paolo Guiotto: Now, this F… Sorry.
25:19:240Paolo Guiotto: this F… is,
25:24:290Paolo Guiotto: Sigma Algebra.
25:27:470Paolo Guiotto: Now, okay, so it's a quite abstract example, let's try to work out together this. Now, what should we do? So far, we have only the definition, so the unique thing we can do is to check if the definition is verified, right?
25:45:80Paolo Guiotto: Okay, so the diffusion means that three, things that must be checked. The empty and full space are inside.
25:53:660Paolo Guiotto: So, do you see… How to do this?
25:57:380Paolo Guiotto: Is it immediate or not?
26:00:850Paolo Guiotto: Empty is in the family. Yes or no? Why?
26:05:960Paolo Guiotto: Yes, of course, but my…
26:08:810Paolo Guiotto: Because really, the condition to be in the family. The condition is at least one of your set, or it's complementary, is countable.
26:18:740Paolo Guiotto: We have… the set is empty.
26:20:900Paolo Guiotto: So at least one of empty, or.
26:24:200Paolo Guiotto: Complementary of empty, the full set, is countable.
26:28:80Paolo Guiotto: Which one is definitely comfortable?
26:31:130Paolo Guiotto: PMT said there is nothing, zero elements, okay? So, clearly… So, solution.
26:41:660Paolo Guiotto: We check.
26:45:930Paolo Guiotto: D.
26:47:750Paolo Guiotto: axions.
26:50:560Paolo Guiotto: Fall.
26:52:370Paolo Guiotto: S, sigma algebra.
26:57:80Paolo Guiotto: Number one, empty…
27:00:490Paolo Guiotto: or X belongs to F. Is that true? Empty belongs to F, if, and only if.
27:09:140Paolo Guiotto: empty, or… empty complementary, which is X, is countable.
27:17:480Paolo Guiotto: And we say that this one is countable.
27:22:620Paolo Guiotto: Of course, as you can understand, also, the same argument works for X.
27:27:180Paolo Guiotto: X belongs to F, if and only if X, or X complementary, which is nothing, is countable.
27:37:520Paolo Guiotto: And in this case, it is this one to the top.
27:41:330Paolo Guiotto: Okay, number two. Suppose that we have a set E that belongs to the family.
27:49:400Paolo Guiotto: And we want to check that also its complementary belongs to the famine. So what do we know?
27:56:600Paolo Guiotto: So this is what we know.
27:59:620Paolo Guiotto: It means that, E, or… E complementary is countable.
28:08:980Paolo Guiotto: One of the two, we don't know which one, okay?
28:12:450Paolo Guiotto: At least one. Now, This is, the hypothesis. We know that this is true.
28:21:300Paolo Guiotto: And what we want, the thesis, is…
28:25:930Paolo Guiotto: Is he complementary in the family?
28:29:310Paolo Guiotto: Well, this is if and or if the set, e-complementary, or it's complementary, so e-complementary, complementary.
28:39:550Paolo Guiotto: Who is this guy?
28:42:290Paolo Guiotto: Eat.
28:44:220Paolo Guiotto: is countable.
28:47:640Paolo Guiotto: As you can see, apart from the order, these two sentences are the same sentence.
28:53:730Paolo Guiotto: Okay, one says either E or E complementary is countable, the other says either E complementary or E is countable. It's the same thing. So, in fact, this is completely equivalent to this, and therefore we get the conclusion.
29:09:520Paolo Guiotto: Now, number 3 is a little bit more complicated.
29:14:100Paolo Guiotto: Because we have to discuss this union. So we take a family of sex, in, in FM.
29:24:50Paolo Guiotto: So this means that each of them fulfills that condition.
29:29:50Paolo Guiotto: So what does it mean? This means that for every N, EN. OR. EN complementary, is, accountable.
29:44:530Paolo Guiotto: What does it mean? That, for example, E1 could be comfortable, E2 complementary could be comfortable, you see? It's not the… for all of them, it is Vienna or the complement, okay?
30:00:360Paolo Guiotto: Each one, at least one of the two, the set of each complementary, is valid.
30:08:110Paolo Guiotto: Okay. That doesn't really see if these,
30:11:780Paolo Guiotto: Not important which one is the countable.
30:15:610Paolo Guiotto: Now, the thesis is, the union of the yen belongs to F.
30:26:570Paolo Guiotto: Okay, this means that, huh?
30:30:380Paolo Guiotto: Either the union belongs, so either the union is comfortable, or… The complementary of the union, is comfortable.
30:44:310Paolo Guiotto: Now, let's, write…
30:46:860Paolo Guiotto: We have this complementary of the union. Maybe it is better if we write… what is this?
30:55:850Paolo Guiotto: The complementary of the union is the… Not me.
31:03:460Paolo Guiotto: Here's the intersection. Do you mind that when we do the complementary, it's like union… they are a complementary of operations. One becomes the other, and vice versa.
31:14:950Paolo Guiotto: Okay, so that's the intersection of the complementaries of the N. Keep in mind these things we will use, especially in the second part, in probability, it happens that we have this kind of operations,
31:28:520Paolo Guiotto: Intersections, unions, complementary.
31:33:170Paolo Guiotto: So, be careful on this. Okay, so, our conclusion means that either the union of the yen or the intersection of the complementaries
31:46:340Paolo Guiotto: of the yen. One of these two sets, at least, is countable.
31:52:820Paolo Guiotto: Now, it is not the median.
31:55:950Paolo Guiotto: to understand how to, reach this conclusion from that assumption. The assumption is.
32:03:750Paolo Guiotto: But in the end, at least one of the two guys, the end or the end componentry, is comfortable.
32:10:670Paolo Guiotto: Okay?
32:12:630Paolo Guiotto: Now, let's try to find a strategy to get that conclusion. The conclusion is that
32:19:910Paolo Guiotto: One of union of the yen of this intersection, the n complementary, is capital.
32:27:190Paolo Guiotto: So, let's say, when do you think that the first one is coming?
32:33:300Paolo Guiotto: What should happen to have this one chemical?
32:38:410Paolo Guiotto: only DRAM resource accountable.
32:41:720Paolo Guiotto: Be better.
32:44:00Paolo Guiotto: all the ER… all the ER must be targeted. Right.
32:49:590Paolo Guiotto: Well, it is a known factor, you know, and maybe… but you can understand intuitive. If you do a comfortable unit, or countable sex, you get comfortable.
32:59:750Paolo Guiotto: Except we just feel like Japan.
33:02:740Paolo Guiotto: But if one of them is uncomfortable, so he's bigger than N,
33:08:540Paolo Guiotto: No? It's not in correspondence with them. It is clear that the union contains that guy, which is not in correspondence with them. The union with CMPF cannot be in correspondence with Dan. Otherwise, also, that's that. So, we understand that we have this, we need that OEM account.
33:28:780Paolo Guiotto: But it is… this is not necessarily true, right? Because the first one is countable, that the second is the complementary.
33:37:180Paolo Guiotto: Okay, now we have to put down an arrow.
33:39:880Paolo Guiotto: So the argument could be, okay, either all the men are comfortable, and then this guy is comfortable.
33:47:380Paolo Guiotto: Or, it's not true that all the Yemi are comfortable, and let's see if the other guy is true.
33:59:640Paolo Guiotto: Well, I'm not supposed to spec si problemo.
34:04:510Paolo Guiotto: I'm gonna look at the results.
34:09:870Paolo Guiotto: So stupid.
34:14:659Paolo Guiotto: All right, what's the question.