AI Assistant
Transcript
00:00:00Paolo Guiotto: Okay.
00:11:30Paolo Guiotto: Good morning.
00:13:540Paolo Guiotto: And,
00:15:200Paolo Guiotto: Before we start, let me renew you the invitation to enroll on Moodle, okay? I see that only 15, 14 of you are enrolled, so…
00:30:170Paolo Guiotto: please enroll there, because it is the place where I will publish on the materials, okay? Also, I'm thinking, too, when I give the assignments, I will not be able to solve all the exercises in class, so maybe someone I will solve in class, but I will publish solutions.
00:48:210Paolo Guiotto: I mean, it'll take some time, I hope, to be able to have time enough to publish all the possible solutions. To the exercise, I leave you.
00:57:400Paolo Guiotto: Yeah, I know, I know. That's a big issue. Hopefully, it should be…
01:08:360Paolo Guiotto: should be solved soon. Okay, so…
01:11:920Paolo Guiotto: Yesterday, we introduced the definition of, abstract measure.
01:22:190Paolo Guiotto: So, it is a function defined on a sigma algebra of sets of a certain number.
01:30:40Paolo Guiotto: environmental space X, with no other specification on the structure of this XM. And this function must verify two apparently simple conditions. Must measure of empty set must be zero.
01:47:50Paolo Guiotto: And the countable additivity property, which says that when you have a disjoint union, the measure of the disjoint union is the sum of the measures.
01:56:760Paolo Guiotto: I want to start with another example of a simple example of measure.
02:06:130Paolo Guiotto: Which will be some… somehow important for some examples in future, so it's a good idea to keep it in mind.
02:14:40Paolo Guiotto: So this second example is the so-called example
02:19:310Paolo Guiotto: Of the, so-called counting measure.
02:30:610Paolo Guiotto: we define, on the set of naturals, but, it could be extended to each X, which is a countable set. So here, X, for this example, will be just a set of naturals, but, we…
02:47:60Paolo Guiotto: can… They cut… And he… It's countable.
02:56:430Paolo Guiotto: set. So, any set which is in correspondence one-to-one with naturals. So, for example, rationals, or more complicated things, but still, with the cardinality of naturals.
03:10:870Paolo Guiotto: And, as a sigma algebra, we take parts of X, so all the possible subsets of naturals.
03:19:340Paolo Guiotto: You know, this is a standard sigma algebra.
03:23:290Paolo Guiotto: And as measure mu, we define the measure of the set E as just the count of the elements of E.
03:33:290Paolo Guiotto: So, we may say, literally, a number of… elements…
03:43:450Paolo Guiotto: of E, with the agreement that if the set E contains infinitely many elements, this number will be plus infinity, okay?
03:53:410Paolo Guiotto: Well, we could write also with a sort of formula, which is, normally is presented in this way. We can sum…
04:03:930Paolo Guiotto: For n going from 0 to plus infinity.
04:08:40Paolo Guiotto: So, this is the range of naturals. For us, naturals, contain zero, okay? So, we, we use this notation, 1NE,
04:21:130Paolo Guiotto: Well, actually, this is the delta, this is the direct delta, delta N is the same, but sometimes we will use this notation 1N.
04:31:280Paolo Guiotto: Perhaps it is a little bit,
04:34:220Paolo Guiotto: better, because it reminds that this quantity is just 1, otherwise it is 0, okay? So it is 1 when N belongs to E, and 0 otherwise.
04:44:770Paolo Guiotto: So this sum counts 1 plus 1 plus 1 for every n that belongs to E, so it counts how many elements are in E. Now, it turns out that this is a proper measure, mu is a measure…
05:02:210Paolo Guiotto: on… on, excerpt.
05:06:740Paolo Guiotto: with the… respect to… Sigma Algebra F.
05:13:920Paolo Guiotto: Now.
05:14:980Paolo Guiotto: It's not a particularly complicated check. The two property, measure of empty, there are no elements, you count zero.
05:23:180Paolo Guiotto: And if you have a disjoint union of set, it is clear that the number of elements of a disjoint union is the sum of the numbers of the elements of each of the components.
05:35:620Paolo Guiotto: Right? Because they are disjoint. And eventually, if you have an infinite measure, you can see that
05:43:00Paolo Guiotto: it is equal, the two sides. Okay, so, I don't want to spend time in verifying this, but it is, it is, a good example, because it is, simple.
05:56:00Paolo Guiotto: And it is mostly used to build counterexamples in future. You will see.
06:03:880Paolo Guiotto: Okay, so let's introduce, just a remark about, notation. So we call, you see these three elements, the space, the sigma algebra, and the measure. This is called, is called,
06:20:140Paolo Guiotto: measure… Sorry.
06:24:450Paolo Guiotto: measure.
06:28:280Paolo Guiotto: space.
06:33:180Paolo Guiotto: In probability, just we spend a second to… we will return on the second part of the course on this, in probability.
06:47:780Paolo Guiotto: The audience?
06:49:480Paolo Guiotto: So, there is an analogous triplet, which is usually denoted by slightly different letters. The set X is denoted with the elector capital omega.
07:01:500Paolo Guiotto: F is F, and the measure mu is, denoted by, normally, what, this sort of applique is B.
07:10:520Paolo Guiotto: is colder.
07:16:970Paolo Guiotto: This time, the probability space.
07:22:960Paolo Guiotto: If, of course, it is a measure of space, If it is.
07:30:70Paolo Guiotto: And measure space, so it means that you have set, sigma algebra, and measure, with the unique additional condition.
07:39:830Paolo Guiotto: with… D.
07:43:650Paolo Guiotto: Say, extra.
07:46:870Paolo Guiotto: condition… that the measure, in this case P, of the full space must be equal to 1.
07:56:40Paolo Guiotto: So, a probability space is nothing but a particular case of a measure space.
08:01:580Paolo Guiotto: Apart from letters, there is a probabilistic flavor in this probabilistic interpretation that says that the elements of the family F, which normally are called the measurable sets in this
08:19:700Paolo Guiotto: It's, set up, so… This is the family… of… Measurable
08:33:549Paolo Guiotto: sex.
08:35:169Paolo Guiotto: So those sets to which we can assign a measure. In the case of the probability space, they are called events, so family…
08:46:500Paolo Guiotto: off.
08:48:910Paolo Guiotto: D… A possible… well, let's say possible… possible.
09:00:130Paolo Guiotto: events.
09:05:500Paolo Guiotto: And, well, the number here, in the measure of theory, the number mu of e is what we call measure E.
09:17:790Paolo Guiotto: All.
09:18:830Paolo Guiotto: E? Well, if you…
09:21:220Paolo Guiotto: If we need to be precise, we should say new measure, because there could be different measures on the same space, as we will see. But if we are just talking about the specific setup, where there is one fixed measure, we just say.
09:34:290Paolo Guiotto: measure of V, while, down here, the analogous, which is the P of E is called the probability
09:46:700Paolo Guiotto: of E with the implicit interpretation that this is the probability that event E happens, okay?
09:55:250Paolo Guiotto: But apart for these terminology notations, they are exactly the same environment.
10:02:600Paolo Guiotto: So what we are going to see and develop for measure spaces.
10:07:250Paolo Guiotto: in 100% can be applied, and will be applied to the probability theory, okay? So when we will define an integral with respect to measure mu, we will have automatically an integral with respect to the probability P that takes another name in probability, but it is still an integral, and so on.
10:26:150Paolo Guiotto: Okay, let's close here this, Parenthesis.
10:31:750Paolo Guiotto: On, on this, terminology.
10:34:770Paolo Guiotto: Okay. Now, other examples of measure spaces. We will see… I will… I think we will start today to give a look to the case of the LeBague measure, which is, we may say, the most important case of a measure space. It's not…
10:52:800Paolo Guiotto: the unique one, because there are important measure space is different from the LeBague case. But as we will see in the second part of the course, it is like if we could do all the probability theory just with the LeBague measure on the interval 01. So, we could use as a prototype of a probability space, the interval 0, 1,
11:16:740Paolo Guiotto: With the traditional measure that we use there.
11:20:140Paolo Guiotto: Yeah, okay, so that's why the LeBague measure is so important.
11:24:940Paolo Guiotto: And, of course, it is very important for analysis. The problem is that to introduce the LeBerg measure is quite complicated, as we will see, very… a lot… there are a lot of technicalities, non-trivial technicalities, so we will not
11:39:850Paolo Guiotto: see all the process that brings to the definition of this measure, but at least we will have an idea of how it works, and we will get some informations of this, okay? So this is going to be the next topic, but before we enter into that discussion.
11:58:510Paolo Guiotto: We will just illustrate a few fundamental factors, general factors, that hold for any measure, okay? So that you can apply to the LeBague measure, but of course, to any other type of measure that you encounter.
12:16:450Paolo Guiotto: In probability, we'll see that.
12:19:40Paolo Guiotto: could be very strange measures. I anticipated that we will see what is the impact, so it gives probability that a certain trajectory happens or not. So, it's a probability defined on an infinite dimensional setup. It's complex, okay?
12:34:820Paolo Guiotto: But nonetheless, it's a measure, and therefore, the general property holds also for that measure. So, let's say…
12:44:320Paolo Guiotto: some… general… Facts, so… about… measures.
12:58:970Paolo Guiotto: Okay, so let's, first,
13:04:130Paolo Guiotto: clarify one point. So, we said that in the definition, the measure mu is countably additive.
13:13:470Paolo Guiotto: as a particular sub-case that must be, say, known, we have defined additivity, okay? So let's do this initial remark.
13:29:440Paolo Guiotto: So, imagine, so let, X, F, new… be a measure of space.
13:42:400Paolo Guiotto: So, the point is that, countable Additivity.
13:51:800Paolo Guiotto: of mu implies what is called the finite.
13:58:970Paolo Guiotto: CBD.
14:01:00Paolo Guiotto: Which is, as you understand, a spatial case. So this means that if you have the measure of a finite disjoint union.
14:09:470Paolo Guiotto: So something like a union for angle from 0 to capital N of certain sets EN.
14:16:450Paolo Guiotto: This is a defined sum, well, the sum… Could be in finite invested.
14:26:800Paolo Guiotto: Okay, because maybe one of the measures has a value plus infinity, so this makes the sound… In final.
14:38:520Paolo Guiotto: But, the… I need adjusted… Why? Because we…
14:46:920Paolo Guiotto: Can, transform this into a… Countable.
14:52:900Paolo Guiotto: this joint union, because you do, E0,
14:58:90Paolo Guiotto: This joint union with D1, this joint union with D2, and so on, until you reach… lleno.