AI Assistant
Transcript
00:06:320Paolo Guiotto: Okay, so… Good morning.
00:11:20Paolo Guiotto: Everyone, so let's start it up here.
00:14:30Paolo Guiotto: We have, seen a few of the principal properties of the auto measure on RM.
00:26:290Paolo Guiotto: So basically, this function, because it won't be a measure, is coherent with the geometry. Measure of the interval turns out to be the size of the interval.
00:40:860Paolo Guiotto: And, moreover, it is, invariant by translations.
00:47:930Paolo Guiotto: It is just countably additive.
00:52:40Paolo Guiotto: But it can be proved that this thing cannot be made a true countable additive measure. Now, I don't want to do the proof of this. If you are curious, you can find
01:05:170Paolo Guiotto: on the notes, the point is that, let's just write just a few, few, few words, is that, for example, in dimension, 1, but you can repeat this construction in any dimension, so take M equal 1.
01:21:710Paolo Guiotto: And it is possible to do the following thing. So, we take the interval, minus 1,
01:30:690Paolo Guiotto: And it is possible to build inside this interval a certain set E,
01:37:450Paolo Guiotto: Which has, the following properties.
01:41:570Paolo Guiotto: It should be a set made of irrational numbers.
01:46:250Paolo Guiotto: Not all the rational numbers.
01:49:200Paolo Guiotto: such that when you take D, translated of this set with durationals that are between 0, 1, I guess, here, no, minus 1, so for Q,
02:04:500Paolo Guiotto: rationale.
02:06:280Paolo Guiotto: And 2 minus 1, 1.
02:09:560Paolo Guiotto: So these are just translations left-right of this set. Now, it turns out that these sets are disjoint. For different Q, you have different sets. So this family is a countable family, because Q is a subset of rational, so it's…
02:27:80Paolo Guiotto: Comfortable, is… Countable.
02:34:500Paolo Guiotto: Family… Of sets,
02:43:90Paolo Guiotto: Of course, since lambda star is defined on every set, also these ones are measurable sets, we can compute the measure.
02:52:270Paolo Guiotto: And since the measure is invariant by translations, they all have the same measures, which is the measure of the set E. But these sets do the following. When you do the union such that
03:05:290Paolo Guiotto: If you do the union of these sets.
03:08:720Paolo Guiotto: verifying the Q in rationals, between minus 1, 1.
03:16:70Paolo Guiotto: So, it turns out that… you see that since you are translating that set at most, plus one right, at least minus one left, so you are in the interval minus 2, so that's, say, is not the coolest part.
03:35:300Paolo Guiotto: What happens is that it can be shown that it contains the interval minus 1v1.
03:42:770Paolo Guiotto: So, in some sense, it is possible to slide… to slice in an countable number of subsets, the set minus 1, in such a way that they are disjoint, and they have all the same measure.
03:58:140Paolo Guiotto: And here it comes the contradiction, because if this is possible, If… this… is possible.
04:09:870Paolo Guiotto: Then, applying lambda star, we would get that the lambda star of the disjoint union, I just write E plus Q over the Q,
04:21:89Paolo Guiotto: Since we have the monotonicity, that set is contained in minus 2, 2, so it will be less than the lambda star of the interval, minus 2, which is an interval, so this is the length, it is 4, and it will be also greater than the lambda star of minus 1, 1.
04:41:30Paolo Guiotto: Which is, the length of the interval, so it is equal to 2.
04:48:340Paolo Guiotto: But on the other side, if, this if, of course, lambda.
04:54:140Paolo Guiotto: Stop.
04:56:320Paolo Guiotto: is countable.
04:59:430Paolo Guiotto: additive.
05:01:560Paolo Guiotto: We could say that the understar of the disjoint union would be the sum
05:08:90Paolo Guiotto: Doesn't matter if we write some of the Q, we can always, put in a comfortable way this Q, no? So we say Q1, Q2, QN.
05:18:110Paolo Guiotto: of the measures of this E plus Q, which, by invariance, by translations, these measures would be all equal to the measure of E.
05:31:810Paolo Guiotto: And that's the contradiction, because we would have 2 less or equal than infinitely many times, the sum of this same number, lambda star E,
05:42:660Paolo Guiotto: Less or equal than 4.
05:44:650Paolo Guiotto: I understand that since you are summing the same constant very, many times, you are only two possibilities. Either that constant is positive, and the sum will be infinite, but this is impossible with this bound, or this quantity is zero.
06:02:420Paolo Guiotto: But then the sum would be zero as well, and this would be impossible with this lower bound.
06:08:940Paolo Guiotto: Okay? Now, the key point is the construction of that setting, which is not intuitive, and even not constructive, because it depends on something like logical arguments that we don't want to enter here. And this,
06:26:780Paolo Guiotto: And this… would be… Impossible.
06:36:850Paolo Guiotto: Because you can say that the sun
06:40:80Paolo Guiotto: over Q of the lambda star E would be equal to plus infinity if lambda star E is positive.
06:49:980Paolo Guiotto: and 0 if lambda star E is 0.
06:54:900Paolo Guiotto: So this example, in a way, shows that that lambda star is not countably additive.
07:02:250Paolo Guiotto: Now, the problem is that lambda star pretends to give a measure to every set.
07:10:20Paolo Guiotto: And, so this allows the existence of these kinds of sets, which shouldn't be there, which shouldn't be considered as measurable sets. Otherwise, we could do such kind of union and have a contradiction.
07:24:950Paolo Guiotto: Okay, so the solution to this issue is not to throw away everything, but it is… well, it is too much to have all sets of RM measurable.
07:38:310Paolo Guiotto: We cannot have an idea of what are, in fact, all the possible subset of the real line. It's a very huge family of sets. But we want to be sure that certain type of sets are in the class of measurable sets. So the idea is
07:56:90Paolo Guiotto: We try to restrict this lambda star to a smaller class of subsets that contains at least some good sets, okay? And now that's the idea that we introduce now, which is the definition of the so-called
08:13:980Paolo Guiotto: Le bags.
08:19:730Paolo Guiotto: class.
08:23:240Paolo Guiotto: Now, the Rebec class is a family of subsets of RN, Which is different.
08:33:409Paolo Guiotto: the symbol MD.
08:36:549Paolo Guiotto: Now, what we want, basically, is that, important classes of sets.
08:44:970Paolo Guiotto: Which are commonly found in applications, are inside this family.
08:51:100Paolo Guiotto: And this class is the class, the class that we want to be there is the class of OpenSets, okay? The open sets, you can see as… let me just refresh for all of you. OpenSet means,
09:05:460Paolo Guiotto: Open… set.
09:08:800Paolo Guiotto: O contained in RM.
09:12:310Paolo Guiotto: is open.
09:15:530Paolo Guiotto: If and only if, for every point.
09:20:770Paolo Guiotto: X of the set O, there exists a hybrid, normally a pole, centered at point X with radius R.
09:34:670Paolo Guiotto: So these are vectors, Y of RM,
09:39:570Paolo Guiotto: Such that the norm Y minus X is less or equal than R. Euclidean norm, the usual norm of R.
09:49:470Paolo Guiotto: Such that this ball… is contained into R.
09:55:330Paolo Guiotto: We usually have the idea that these sets are subsets without the boundary.
10:01:540Paolo Guiotto: So, at the inside of some regions.
10:04:890Paolo Guiotto: without the skin. Because if you take a point here, you can say there is a small ball entirely contained. Take a point here, maybe the ball is bigger, but there's not a general radius that works for every point. But no matter how you take the point, there is always a little ball until you can turn it into the set.
10:23:970Paolo Guiotto: Now, for technical reasons, we take also the empty set as an open set. The property does not make sense, so empty is…
10:34:740Paolo Guiotto: assumed that…
10:39:870Paolo Guiotto: open…
10:43:480Paolo Guiotto: Just to be clear why these are so common sets, I remind you that, for example, if you take a set like this, the set of points X of RM,
10:56:70Paolo Guiotto: Normally, a common, very common way to define subsets is through a certain number of equations, inequalities, things like this, no? Now, if you take an inequality, like G , strictly less than 0,
11:11:620Paolo Guiotto: you can write any kind of strict inequalities in this form, or strictly greater than zero. With G, a continuous function.
11:21:330Paolo Guiotto: we…
11:23:460Paolo Guiotto: G, a continuous function on RM, so the set of points that verify this inequality, no, something like X squared plus Y squared plus Z squared is still less than 1. It is the interior of the ball centered in the origin and radius 1.
11:41:870Paolo Guiotto: So, this kind of set is, open.
11:48:290Paolo Guiotto: And you can have also, if you have any number of constraints, you know, G1, G2, G3, GK, less than 0, each less than 0, you have still an open set. So this is a very common family.
12:02:520Paolo Guiotto: Then there are also common feminists, like closed sets, in that case, we have a large inequality, so less or equal, or a inequality, no? G equals 0.
12:13:710Paolo Guiotto: Okay? But as you will see, they will be, also there. So our family at the end will contain open and both closed sets, okay? The two.
12:24:80Paolo Guiotto: Now, what we want is that at least this family is contained, and also the intervals, but you will see they are there. So, the starting point is that we want that open sets are there.
12:40:70Paolo Guiotto: and sets which are more or less very similar to OpenSet, similar in the sense of measure. So the idea behind this definition is that we can say that we take subsets E,
12:56:630Paolo Guiotto: Sorry.
13:01:00Paolo Guiotto: subset of R.
13:03:660Paolo Guiotto: M… Such that, well, we can, in words, we would say, E… ease.
13:11:990Paolo Guiotto: Well… Approximated… by… and open.
13:23:420Paolo Guiotto: set.
13:24:560Paolo Guiotto: well approximated in the sense of measure. What does it mean? It means that, let's do a figure somewhere, I will do down here the figure. So, my set E is the set I want in the class.
13:38:700Paolo Guiotto: The set is in the class if I can approximate with an open set.
13:44:940Paolo Guiotto: from above, so there is an open set that contains E, so let's imagine this open set, something like, that must contain this, E, so it will be bigger.
13:56:600Paolo Guiotto: then E…
14:01:200Paolo Guiotto: But such that the measure, the outer measure, in fact, of the excess, so what is in this open O, and it is not in E, so let's say this part here.
14:17:690Paolo Guiotto: The measure of this success… Can be made arbitrarily small.
14:25:10Paolo Guiotto: You see? So it is like if I'm saying there is an open set such that it contains E, but in such a way that I can choose that open set in such a way that the measure of the excess, what is outside of E, is small. Now, we write formally this in this way.
14:42:400Paolo Guiotto: So, for every epsilon that's positive, that we represent how much we want.
14:47:240Paolo Guiotto: the excess be smaller. There exists an open set, O epsilon, open.
14:55:570Paolo Guiotto: Such that, O, epsilon contains E,
15:02:230Paolo Guiotto: and it's apparently quite long to write, the outer measure of the accessor, so what is in the open and it is not in E, is small.
15:16:30Paolo Guiotto: So the formal definition is pretty long. The intuitive idea is that you are in the family MD if you are approximating from above, let's say, by some open set, in a way that the approximation is pretty good, in which way?
15:35:310Paolo Guiotto: The ATA is dead. We want that, P-measure of what is outside this wall, okay?
15:41:830Paolo Guiotto: Now, this definition, let's see…
15:45:250Paolo Guiotto: I… we don't really need to work with this definition, because in a few seconds, we will understand that this family contains basically whatever we can imagine, and we can use in applications. But let's just see a few examples that
16:02:640Paolo Guiotto: more or less easily comes from this definition, okay?
16:07:380Paolo Guiotto: So… Let's say, facts.
16:15:40Paolo Guiotto: The final goal will be that this family is a sigma algebra.
16:19:590Paolo Guiotto: And this family contains intervals and open sets and closing sets as well, okay? Let's start with the first simple check, is that this family contains all the open sets. That is basically trivial.
16:37:490Paolo Guiotto: So, if old is… open, Then, O belongs to the family MD.
16:49:730Paolo Guiotto: So the family contains all the open sets.
16:52:970Paolo Guiotto: In fact, if you look at the definition.
16:56:90Paolo Guiotto: Imagine that our set, E, that we have here is just an open set.
17:03:350Paolo Guiotto: So, you have your set,
17:07:250Paolo Guiotto: Which is the open set O.
17:10:859Paolo Guiotto: And you want to find an open set that contains O in such a way that the measure of the excess is small.
17:20:520Paolo Guiotto: But in this case, you can take it just as O epsilon, the same obe.
17:27:260Paolo Guiotto: It is open.
17:31:740Paolo Guiotto: And what is the excess? O epsilon minus O.
17:35:580Paolo Guiotto: is nothing.
17:37:970Paolo Guiotto: So, the measure of the excess…
17:44:640Paolo Guiotto: Is the outer measure of nothing that we already said, that it is equal to zero.
17:51:30Paolo Guiotto: So in this case, it can be made less than epsilon, whatever is epsilon.
17:57:940Paolo Guiotto: So, immediately, we have that open sets are in the family package.
18:03:800Paolo Guiotto: Let's verify, for example, that intervals are also in the family. Intervals, we always mean intervals, rectangles with the boundary, so technically speaking, they are not open sets, because of the boundary.
18:16:880Paolo Guiotto: But, in any case, I belongs to MD… for every i interval.
18:30:130Paolo Guiotto: Well, we can understand by doing a sort of graphical proof, and let's do this proof in a low dimension, so we don't need to do it in general to get a message. So let's do in dimension M equal… sorry…
18:49:620Paolo Guiotto: I use letter D, I'm doing a mess.
18:53:850Paolo Guiotto: So this… I see, there is… so let's use the letter D, sorry. Is that I want… I wanted to use a different… I don't want to use the letter N, because it will be the index, so it will be associated also with the idea of an index. Letter N stands…
19:10:790Paolo Guiotto: Not so nice. We should use something like MK, but K is also used as an index.
19:16:640Paolo Guiotto: The D is not a good letter, because we have a new together, there will be PX, so it can… can give some confusion, so I don't know what,
19:28:860Paolo Guiotto: So let's say, let's take letter K. Let's put the letter K everywhere, and… So, M, K, R, K…
19:37:670Paolo Guiotto: So this is RK… This is, archaic…
19:44:710Paolo Guiotto: I will try to use… to do not use letter K as an index for sequences, but…
19:52:40Paolo Guiotto: So this is MK… sorry for the…
19:59:470Paolo Guiotto: Okay, so here we have MK. So let's do the proof only for the case.
20:05:710Paolo Guiotto: K equals 2. Proof. I don't want to do, really, proofs, because to write things, it demands some little K. So take a rectangle i contended into R2, so
20:21:50Paolo Guiotto: interval. In this case, it is a rectangle.
20:26:150Paolo Guiotto: Now, this is our set. We want to show that it is a measurable set, so it belongs to that class. We need to find an open set that contains this i in such a way that the measure of the excess is more.
20:38:440Paolo Guiotto: Well, we could take a slight enlargement. Eye itself cannot be taken because it's not open, but they can take a slight enlargement of our eye.
20:49:880Paolo Guiotto: Like… the boundary here.
20:56:10Paolo Guiotto: So, if I, just to write letters, if I is, interval AB, times CD.
21:06:920Paolo Guiotto: this, red set that I call O epsilon could be…
21:14:100Paolo Guiotto: I don't know. I have to slightly enlarge the base, so I will take something like A minus epsilon, B plus epsilon.
21:22:740Paolo Guiotto: across something like a C minus epsilon.
21:27:430Paolo Guiotto: D plus X, you know.
21:29:530Paolo Guiotto: Now, I want to take open, so we have to take open intervals here.
21:35:280Paolo Guiotto: So take this, open it over here, open it over there, Now, this is open.
21:44:10Paolo Guiotto: Because there are not the boundary points. Of course, it contains our eye…
21:52:460Paolo Guiotto: And we have to measure the excess. The excess is this part I now color in red, yeah.
22:06:980Paolo Guiotto: Okay, so you can understand that easily the measure.
22:12:830Paolo Guiotto: of the excess, so lambda star of O epsilon minus i will be, something like, I don't know, the area of this rectangle down here is, what is, A minus B plus 2 epsilon.
22:31:550Paolo Guiotto: It will be A… B minus A plus 2 epsilon, this is the length, times epsilon.
22:40:440Paolo Guiotto: So this is the size of this rectangle down here. There is another rectangle above, so that's 2, so I have also this one.
22:52:600Paolo Guiotto: And then I have these two rectangles.
22:57:50Paolo Guiotto: They have, base epsilon, so plus 2 times epsilon, and,
23:02:640Paolo Guiotto: The 8 is just as D minus C.
23:08:50Paolo Guiotto: So, in any case, you'll see that this quantity is more when epsilon is more, okay?
23:14:740Paolo Guiotto: So, let's say that it is something like constant times epsilon, and therefore you can understand it can be made as small as you like, provided the epsilon is small.
23:26:880Paolo Guiotto: So this family contains intervals.
23:31:100Paolo Guiotto: What else, well, what is not immediately but important class of sets? Also, if…
23:44:170Paolo Guiotto: So, I numbered 1… 2… 3…
23:53:240Paolo Guiotto: If the outer measure of set N is 0,
23:59:830Paolo Guiotto: As we will see, Measure Zero Set play an important role, huh?
24:06:100Paolo Guiotto: And they are… they are not necessarily empty sets, so the empty set has measure zero. That's a trivial set. But it's plenty of sets where the outer measure is 0. I will show some in a moment.
24:22:460Paolo Guiotto: Then, automatically, this set belongs to the class.
24:26:260Paolo Guiotto: MD.
24:28:900Paolo Guiotto: Well, to prove this, is,
24:33:360Paolo Guiotto: Is, a bit, more complicated.
24:37:470Paolo Guiotto: So, because, we don't know exactly how it's made a Measure Zero set.
24:44:490Paolo Guiotto: The only thing we know about the measure of zero set, this is something that could be useful to remind, is that for every set, we know that the lambda star n is that infamom.
24:57:580Paolo Guiotto: If of sum of sizes of ion, where the ion
25:05:480Paolo Guiotto: COGA, power set, in this case, capital N, and they are intervals.
25:13:450Paolo Guiotto: So, if this quantity is zero, I'm saying that the infamum, the best lower bound of the set made of these numbers, is zero.
25:22:850Paolo Guiotto: So it means that no matter How you go above…
25:27:930Paolo Guiotto: of the infamum, you're reminded the key properties of an infamum. We're reminded of these properties, I think, about here.
25:38:540Paolo Guiotto: No? When you have infinone, this number, the big number, which is greater than the infomer, there is always an element of the set below the target. Our set here is made of…
25:52:720Paolo Guiotto: These numbers, these sams, the set is not made by a union, by its profits, it's made by consensus.
26:00:910Paolo Guiotto: So, if B0, it means that whenever I take a positive number, let's say epsilon.
26:08:290Paolo Guiotto: There exists some element of the set down here.
26:12:970Paolo Guiotto: So it means there is just a covering made of intervals of the satan, so there exists
26:19:770Paolo Guiotto: a covering family, IN.
26:23:870Paolo Guiotto: Since it depends on epsilon, let's put an epsilon somewhere.
26:27:720Paolo Guiotto: Such that this family is a covering for our set N.
26:34:670Paolo Guiotto: And the sum of the sizes of these intervals is smaller than…
26:42:720Paolo Guiotto: This number, epsilon, is more than epsilon.
26:46:840Paolo Guiotto: So, a measures you set can be characterized in this way.
26:51:390Paolo Guiotto: for every epsilon positive, I can always cover with intervals in such a way that the total size of this interval is more. Okay, maybe there are infinitely many intervals with a positive size.
27:04:480Paolo Guiotto: But, this size, can be made small.
27:09:200Paolo Guiotto: Okay.
27:10:630Paolo Guiotto: So, we don't know how this set is made, but we know that it is contained into these guys.
27:17:990Paolo Guiotto: Now, you may imagine that since these are something like closed intervals, I cannot use…
27:24:860Paolo Guiotto: He's sad to say that this is an open set.
27:28:830Paolo Guiotto: Well, there is some property that perhaps… let's see if you remind. When you do a union of mobile sets.
27:43:830Paolo Guiotto: have an open set. So if they want to see the topic.
27:48:240Paolo Guiotto: However, it's not a big problem. I can…
27:51:180Paolo Guiotto: So, I cannot take out the points, because if I take any given points, I want to reserve and do not reserve this, huh? But I can enlarge this.
28:02:980Paolo Guiotto: Now, enlarging each of them. This is also to be careful, because the eye is pretty many. So if I enlarge each of them by a little quantity like the eye of enlarge material, not a huge one.
28:14:570Paolo Guiotto: So this enlargement has to be made in a… with a little tricky way. So let's say that they are intervals
28:23:100Paolo Guiotto: So, let's take for… for the purpose of the proof, let's keep… I'm sorry, I'm still using the letter D.
28:30:780Paolo Guiotto: Why this should be K.
28:33:590Paolo Guiotto: Let's say that for the purpose of this argument, let's do it for k equals 1, in such a way that each of these intervals is an interval of type AN
28:46:830Paolo Guiotto: BN, okay?
28:50:760Paolo Guiotto: Now, I want to enlarge a bit this interval. This means, take JN, epsilon, made like Open.
29:04:960Paolo Guiotto: a yen minus something.
29:08:190Paolo Guiotto: I cannot put a fixed number, because if I enlarge of a fixed constant, I assign a constant positive quantity to the measure, and when I sum up, I get infinity. So I have to put something like an epsilon n.
29:23:240Paolo Guiotto: that in the idea will be a quantity that will be chosen small when n increases, BN plus epsilon n. Now I'm sure that this contains this one, so the union of the I, of the J
29:42:280Paolo Guiotto: N, epsilon. This is… they are open, so the union is open.
29:50:230Paolo Guiotto: And that's my candidate, O epsilon.
29:53:860Paolo Guiotto: It contains the union of DIN, epsilon.
29:59:40Paolo Guiotto: that contains our I, so it is still a covering of I. It contains I,
30:09:650Paolo Guiotto: We have not yet decided about this epsilon n.
30:13:720Paolo Guiotto: We will do in a momentum. So far, we can take whatever. The unit condition is that epsilon n must be, of course, positive numbers. It's the way that we enlarge the data.
30:24:380Paolo Guiotto: And what is the measure of the excess?
30:28:440Paolo Guiotto: So when I do the lambda star of, of the excess, O epsilon minus N,
30:38:760Paolo Guiotto: I want to show that this quantity can be made smaller.
30:44:90Paolo Guiotto: Okay, so what can you do?
30:49:210Paolo Guiotto: Sorry.
30:50:830Paolo Guiotto: was not high, but N. I'm doing a little bit of mess, I apologize for this, but our set is N, okay? This is a covalent of N, such that the sum of the sizes is smaller. Okay, so because the sum of this is small.
31:07:780Paolo Guiotto: Once the model said that there was the sum of this can be made smaller, because this would provide a boundary.
31:15:510Paolo Guiotto: I am too bound with this. The idea should be that this guy is itself too small, because it's, just a slight extension of his front that others move down.
31:25:280Paolo Guiotto: Now, what they do is, just throw away this minus sign, it doesn't, buggle, it doesn't,
31:31:460Paolo Guiotto: hack too much to show that this is small. I am computing this measure. Now, this sector, O epsilon minus subt, is less than Y epsilon is 4, okay, because I'm taking out n. So, if I say that this is contained in O epsilon.
31:47:280Paolo Guiotto: What is the relation between these two measures, O epsilon and the measure of O epsilon minus N?
31:54:850Paolo Guiotto: Of course, this set is bigger, the measure will be bigger, because of monotonicity. So I say, less required.
32:02:330Paolo Guiotto: Okay, now, O epsilon is a union, so lambda star
32:06:890Paolo Guiotto: of that union of J and epsilon.
32:11:120Paolo Guiotto: Now, this won't necessarily be a disjoint union, but who cares? The idea is that we are going to prove that this is smaller, so I just use the sub-additivity, which is always working for the star, so this is by sub…
32:28:730Paolo Guiotto: Additivity. I'm not using the count of additivity, which we don't know. So this will be less than the lambda star, sum of the lambda star of these J and epsilon.
32:40:390Paolo Guiotto: And now, what should be the size of this JN? You see, JN is… this interval is open. Well, I can say that this is,
32:51:50Paolo Guiotto: Less or equal than the sum of the lambda star
32:54:890Paolo Guiotto: of… you take a slightly bigger interval by handing the end points, so you take AN minus epsilon n, BN plus epsilon n.
33:09:10Paolo Guiotto: So this is containing our JN, because JN is the same thing without the endpoints.
33:14:920Paolo Guiotto: So the unique difference is the two endpoints I added, but I am still increasing, so the measure is increasing. Now, this quantity is what? This is the measure of an interval, so final point minus initial point, and this is BN.
33:31:200Paolo Guiotto: minus AN plus 2 epsilon n, when you do the calculation. So some of these…
33:39:250Paolo Guiotto: But the sum of BN minus AN, BN minus AN is what? Is the length of this, so the size of IN epsilon is exactly BN minus AN.
33:52:40Paolo Guiotto: So that number here is the size of IN epsilon plus 2 times epsilon n.
34:01:890Paolo Guiotto: So, our sum is the sum of these sizes
34:07:480Paolo Guiotto: I and epsilon, plus 2 times the sum of the epsilon n.
34:14:389Paolo Guiotto: Now, by construction, we know that the sum of the sizes of this I and epsilon, they come from this property, is small, because they were covering a measure of zero set. So this number is less or equal than epsilon.
34:30:580Paolo Guiotto: So we just now need to make all this less or equal than epsilon to have that everything is less than 2 epsilon.
34:38:860Paolo Guiotto: Okay, you can put one half everywhere, and you get your epsilon. Now, this means that we have to choose now this epsilon n in such a way that the sum is small.
34:51:620Paolo Guiotto: How can I do that?
34:53:449Paolo Guiotto: Well, there is a standard trick, which is you take any epsilon n, which has a finite sum.
35:00:980Paolo Guiotto: Okay, for example, take 1 over 2 to the n, or 1 over n squared, for example.
35:08:160Paolo Guiotto: The sum of this epsilon n is just 1.
35:11:760Paolo Guiotto: The sum of fraction 1 half, 1 fourth, 1 over 8, and so on, is just 1, and you multiply by your epsilon.
35:20:760Paolo Guiotto: The constant. So when you do the sum of the epsilon n here, you have the sum of fractions 1 over 2 to the n times epsilon, that's 1,
35:31:230Paolo Guiotto: And you are done.
35:32:820Paolo Guiotto: Okay? So it means that if we choose this epsilon n.
35:37:120Paolo Guiotto: We have found, at the end.
35:40:990Paolo Guiotto: an open set, which is made of the union of the J and epsilon, where the J and epsilon at these intervals, such that the measure of the excess is small, and this means that the set is in the family.
35:55:390Paolo Guiotto: So this means that N belongs to MK.
36:01:470Paolo Guiotto: So, you see, it's not easy. It's plenty of this kind of technicalities improving, so I will not prove anything more than this. But this was just to give you a flavor of the kind of…
36:15:790Paolo Guiotto: of technicalities that are behind these, definitions. So now we know that the family contains measure zero set, and now it's, I don't prove this, maybe if you are interested, you can read on the note, or you can try to prove by yourself.
36:32:890Paolo Guiotto: It's not difficult, you can… maybe I will give you the starting point, and then you try to finish. It is this property that says that, if,
36:47:840Paolo Guiotto: A belongs to the family.
36:50:940Paolo Guiotto: And… N is a measurable set with measure equals 0, so lambda star n is equal to 0.
37:01:650Paolo Guiotto: Ben… When you take their union, E union N, this will belong to the FAM.
37:08:990Paolo Guiotto: And actually, also, E minus N,
37:13:350Paolo Guiotto: Both these belong to the family.
37:15:990Paolo Guiotto: So you can add or subtract a measure of zero setter, or measurable setter, and you still have a measurable set.
37:23:450Paolo Guiotto: It means that it doesn't matter what happens on a measure zero set of bonds.
37:30:510Paolo Guiotto: This does not change the fact that this belongs to the family. If you are, you stay skin in the family. If you eliminate, you still stay in the family.
37:41:00Paolo Guiotto: Okay.
37:42:290Paolo Guiotto: now… Suppose that you want to prove the first one.
37:49:60Paolo Guiotto: So, let's see, a sketch, huh?
37:58:60Paolo Guiotto: of the… is, nothing more than working with the definition.
38:10:410Paolo Guiotto: So, you know that E is in the family, so this means that what you know is that for every epsilon positive, there exists an open Opsilon open
38:24:830Paolo Guiotto: such that the measure of the excess O epsilon minus E is smaller, and of course, this O epsilon contains E.
38:41:840Paolo Guiotto: Now, you want to show that if you add a measure zero set, So… the goal…
38:52:580Paolo Guiotto: is that, EUNION N belongs to the family MK.
39:00:720Paolo Guiotto: So the same property holds.
39:04:920Paolo Guiotto: So the question is, clearly, how do we find a set O epsilon?
39:11:320Paolo Guiotto: that contains E union N, and in such a way that the measure of the excess is more.
39:18:880Paolo Guiotto: Now, actually, we just come to prove that also N is in the family, right? We just proved. So, we know that there exists also an open set, say, O epsilon tilde, still open, that contains
39:35:370Paolo Guiotto: the node set, N, and such that the measure of the excess epsilon minus N is more.
39:44:650Paolo Guiotto: Okay, so we have, our E,
39:48:950Paolo Guiotto: which is contained into this O epsilon.
39:54:300Paolo Guiotto: Then we add the NN,
39:56:710Paolo Guiotto: which should be a measure zero set, so not imagine something like this with ordinary measure, which is itself contended into another open set, now O epsilon tilde.
40:08:970Paolo Guiotto: Now, as you can imagine, if we do the union of the two.
40:13:400Paolo Guiotto: O epsilon, union, O epsilon tilde.
40:17:920Paolo Guiotto: It's open, because it is the union of two open sets, and it contains the union.
40:24:140Paolo Guiotto: he's open.
40:27:490Paolo Guiotto: And… contains E union N. Now, the problem is to show that when we do the excess of this.
40:38:260Paolo Guiotto: respect to the union.
40:41:180Paolo Guiotto: So, prove that.
40:48:300Paolo Guiotto: If you… take lambda star of this O epsilon union.
40:53:760Paolo Guiotto: O epsilon tilde minus E union N,
40:59:910Paolo Guiotto: This measure is small. Of course, you have to use the facts that you have above.
41:07:190Paolo Guiotto: Maybe it will be to epsilon, I don't know what is the… but let's say constant epsilon, yeah?
41:16:610Paolo Guiotto: You have to try to… Reduce this calculation to these quantities, this and that one.
41:25:900Paolo Guiotto: Okay? So, try to do this exercise.
41:31:330Paolo Guiotto: Okay, now, with this kind of war, which is, I say that…
41:38:60Paolo Guiotto: very technical, but not extremely difficult. There are no particular hard theories behind. It can be proved, this theorem.
41:51:710Paolo Guiotto: Now, this family, MK, is a sigma algebra.
42:00:650Paolo Guiotto: off.
42:02:480Paolo Guiotto: subsets, huh?
42:06:520Paolo Guiotto: of Arcade.
42:09:640Paolo Guiotto: So, it means that it contains empty and the full space. This is… can be checked in a straightforward way, because if you want, both are open, so you know that open sets are in the family, you know.
42:25:40Paolo Guiotto: that this is true. What is, let's say, less trivial is to prove that if a set is in the family, also it's complementary belongs to the family.
42:37:00Paolo Guiotto: And the family is closed for countable unions of sets. So if you take…
42:43:870Paolo Guiotto: In the family, accountable family of sets of this class, you do the union, you still are in that class.
42:51:40Paolo Guiotto: So, let's… let's remind here the few facts.
42:59:250Paolo Guiotto: MK.
43:01:360Paolo Guiotto: contains, intervals.
43:06:910Paolo Guiotto: So let's put in parent intervals.
43:09:800Paolo Guiotto: Open… close the Sets, and many other, okay?
43:19:290Paolo Guiotto: There are also sets which are… it's not necessarily true that the set, either it is open, or it is closed. It can… it can be none of the two.
43:29:60Paolo Guiotto: But let's say that once you know that there are open sets, there are other types of sets that you can build from open sets. For example, in general, when you do the union of open sets, you still have an open set, but when you do the intersection, this is no more true.
43:45:20Paolo Guiotto: Okay?
43:46:230Paolo Guiotto: In any case, doing intersections of open sets will give a set which is in that family.
43:53:110Paolo Guiotto: Of course, countable intersections, and so on.
43:57:60Paolo Guiotto: This is to say that this family is actually so large that contains every set we may wish for any kind of application.
44:06:790Paolo Guiotto: And moreover, when we think to lambda star.
44:10:600Paolo Guiotto: Restricted, so limited to this class of subsets.
44:16:110Paolo Guiotto: And… Lambda? No… It's just a good time. So…
44:28:720Paolo Guiotto: I'm sorry the electricity has gone, so we take the break now.
44:34:830Paolo Guiotto: Bye.
44:41:270Paolo Guiotto: So, lambda star Restricted to this family.
44:47:480Paolo Guiotto: MK, which is now a sigma algebra.
44:51:140Paolo Guiotto: With values in 0 plus infinity.
44:55:600Paolo Guiotto: Easy.
44:57:510Paolo Guiotto: It may show.
45:00:580Paolo Guiotto: So, turns out that it fulfills, in particular, the accountability.
45:05:410Paolo Guiotto: Now, MK is called, the LeBank class, we already said, Little bag.
45:16:240Paolo Guiotto: Glass, huh?
45:18:180Paolo Guiotto: And, lambda star K, When restricted to the family MK,
45:25:850Paolo Guiotto: It is denoted just by lambda K,
45:29:330Paolo Guiotto: If the dimension is clear, lambda.
45:34:240Paolo Guiotto: Okay, we don't need to specify the dimensions. And this is called the LeBague measure.
45:42:530Paolo Guiotto: And it will be the standard reference measure we will consider on RK.
45:50:160Paolo Guiotto: Well, let's say that that's it on the LeBague measure. We cannot do… Too much.
45:58:620Paolo Guiotto: For the moment, with this measure, because we don't have a practical way to compute measures of sets.
46:06:280Paolo Guiotto: Nonetheless, there are a few exercises that you can try.
46:10:640Paolo Guiotto: So… Do these exercises, the 2, 3, 1, the number 2,
46:17:520Paolo Guiotto: Which is on a very nice decanto set. It's not particularly complicated, the exercise 2.
46:25:660Paolo Guiotto: There is a variation I have not written here, which is the exercise 3 and 4. If you want, you can do… but these are the exercises for which I will write the solution that I will publish, and maybe for a few of them, I will do the solution in class.
46:43:140Paolo Guiotto: Not tomorrow, maybe we will, we will wait until Wednesday.
46:49:770Paolo Guiotto: And, so also number 5, 6,
46:53:680Paolo Guiotto: The last two, seven and eight, are a bit more difficult.
46:58:800Paolo Guiotto: Okay? Tricky. Especially D7.
47:03:480Paolo Guiotto: is quite tricky. I found a phone, on,
47:09:580Paolo Guiotto: some exercise, that was for a PhD course.
47:15:480Paolo Guiotto: It's… in fact, it is elementary. It's just, thinking about, sets, intersections, and so on, but…
47:23:730Paolo Guiotto: It took a little bit of time to find out the solution, at least to me, so I think it's not so easy.
47:31:710Paolo Guiotto: You can… you should try, definitely, to do that.
47:36:390Paolo Guiotto: Okay.
47:38:60Paolo Guiotto: let's say that that's all for the moment for the LeBague measure, and let's return to the general discussion, opening the next topic, which is, which is measurable
47:55:740Paolo Guiotto: functions.
48:01:660Paolo Guiotto: So what we are going to do with the measure is to construct an integral.
48:06:620Paolo Guiotto: So we need to integrate functions, and we have to specify which kind of functions are we going to consider, and that's the class of measurable functions, okay?
48:18:510Paolo Guiotto: Now, there are different, many different equivalent, a little bit more general definitions. I now will give here a definition, because maybe this could be,
48:35:110Paolo Guiotto: more immediate for the checks, but it's not the most general, or maybe you can find somewhere else other definitions. So, we have…
48:46:240Paolo Guiotto: space, measure space, X, F, new.
48:52:500Paolo Guiotto: be, measure… space. Another point with definitions of measurable functions is that
49:03:10Paolo Guiotto: Sometimes it is convenient to consider functions that can take values even plus infinity minus infinity.
49:12:880Paolo Guiotto: This to avoid, in certain cases, well, to have that in certain cases where the function is infinite, let's say.
49:23:700Paolo Guiotto: you are still in the class of acceptable functions. So I don't want to be, extremely formal here, but let's say that a function f
49:39:350Paolo Guiotto: defined on some set E.
49:42:430Paolo Guiotto: contained in X with values in R,
49:46:820Paolo Guiotto: I repeat, sometimes you will see…
49:49:970Paolo Guiotto: From minus infinity plus infinity, in the sense that you consider you allow also the possibility that the function can have value equal to plus infinity, equal to minus infinity.
50:01:990Paolo Guiotto: Of course, as you will see, for interesting situations, this may happen, but on a measure zero set. So, the set of points where this can happen for integral functions, we will see, is a negligible set. So, in fact.
50:18:790Paolo Guiotto: You can always do it in such a way that you can throw away this possibility, but…
50:23:610Paolo Guiotto: Let's say that, so this function is, Called… measurable function.
50:41:400Paolo Guiotto: If the following property holds, And the property is the set where F belongs to I, Yeah, I is…
50:53:950Paolo Guiotto: Content in R is an interval.
50:58:600Paolo Guiotto: So, any interval of any nature.
51:02:310Paolo Guiotto: Okay? So with the end points, without the end points, with one endpoints and not the other, etc.
51:09:590Paolo Guiotto: any interval. Now, what does it mean, this sector?
51:14:550Paolo Guiotto: It is better to use this kind of notations, even if they are a little bit improper.
51:21:800Paolo Guiotto: This is a set of wash. Not really a set is a set of elements of something, no?
51:27:400Paolo Guiotto: So, what do you mean? It is this, what this notation could be?
51:32:680Paolo Guiotto: So now this is a set of points, X, huh?
51:35:920Paolo Guiotto: Precisely, it is.
51:38:940Paolo Guiotto: set of points X. Set that belongs to the domain of the function, which is here the set E.
51:45:280Paolo Guiotto: such that f of x belongs to I.
51:50:900Paolo Guiotto: Of course, this is the formal definition of visa.
51:55:710Paolo Guiotto: But maybe this is a more immediate.
51:58:90Paolo Guiotto: is the set when F, values of F belong to this range i. For example, imagine that I is the interval from A to B. It means F between A and B. So, F larger than A and less than B.
52:13:00Paolo Guiotto: It is a little bit more immediate, so that's why we often use these kind of notations.
52:19:630Paolo Guiotto: But formally, it is the other set. Now, the requirement is that this set belongs
52:26:800Paolo Guiotto: to the class of F sets, so it is measurable set.
52:32:590Paolo Guiotto: Whatever is I interval.
52:36:190Paolo Guiotto: That's the key property.
52:44:600Paolo Guiotto: Now… This is the definition.
52:48:340Paolo Guiotto: Some of you could notice that, actually, for this definition, the measure mu is not needed.
52:58:990Paolo Guiotto: Because it's just a property that he bought.
53:03:420Paolo Guiotto: Proxim device. But the point is that we are requiring that the system set of points belongs to the familiar.
53:10:250Paolo Guiotto: And there is nothing because, for example, don't say anything about them, you know what you said.
53:15:830Paolo Guiotto: Okay? So, strictly speaking, We don't need to have a measure of the farm.
53:22:690Paolo Guiotto: Okay, this is a definition that depends on the function, and on the smart about these things.
53:30:160Paolo Guiotto: after that, we always have a measure if we have a demand, so we always have things set up, okay? But for what we are going to see.
53:40:100Paolo Guiotto: At least for a part of properties of measure or functions, the fact that there is a measure is not important.
53:47:510Paolo Guiotto: a part of facts. Now… Let's see…
53:53:520Paolo Guiotto: examples of measurable functions. So let's start getting familiar with the definition and with this class. The idea that you will have, at the end.
54:06:990Paolo Guiotto: exactly as we have seen, for example, in the case of the Lebec measure, we have seen that the Lebec class at the end contains basically everything we could imagine, and it is difficult to imagine a set which is not there, not there. In fact, there are sets which are not in the Lebec class.
54:26:950Paolo Guiotto: No, sorry, they are here in today class. There are sets which are not in the Lebec class, because the set I mentioned at the beginning of
54:36:430Paolo Guiotto: previous hour.
54:39:140Paolo Guiotto: the set that makes, the measure, the outer measure not countably additive, that set cannot be in the Lebagic class.
54:50:290Paolo Guiotto: Otherwise, the translated of the set would be in the Lebec class, and we would have a contradiction with countable additivity.
54:58:480Paolo Guiotto: So that class is certainly not the class of all the subsets.
55:04:450Paolo Guiotto: But it is not easy to identify who are the bed sets which are not in the Lebec class. So we normally think that, basically, for practical purposes, everything is measurable, okay? Exactly here.
55:22:260Paolo Guiotto: You will see that the class of measurable functions is huge.
55:28:350Paolo Guiotto: is much larger than what we can imagine, because in fact, it is at least in correspondence with the class of measurable sets. And this is just the first example. Example 1.
55:41:870Paolo Guiotto: Let's say this is one star example, very easy, very important.
55:46:450Paolo Guiotto: It's a very elementary type of function, which is the indicator. So, indicator of,
55:56:110Paolo Guiotto: of the set E.
55:58:480Paolo Guiotto: is the function which is defined, is the usual… is always the direct delta, basically, but here we emphasize the fact that we do not look at this as a function of the set E, but rather as a function of point X. It is the same of the notation we used was delta X E.
56:17:750Paolo Guiotto: And in this case, as function of set E, X fixed, this is a measure.
56:22:610Paolo Guiotto: Here, we forget this thing because we focus on how this is a function of X point. E is fixed. So, it is always the same thing. It is 1,
56:34:710Paolo Guiotto: If X belongs to E, it is 0. If X does not belong to E.
56:40:410Paolo Guiotto: Now, this function is measurable, The indicator, this is called indicator.
56:51:470Paolo Guiotto: function.
56:53:960Paolo Guiotto: of E.
56:56:150Paolo Guiotto: Or some other time is called the unit.
57:01:740Paolo Guiotto: Fun is done.
57:05:290Paolo Guiotto: So this function is measurable.
57:10:400Paolo Guiotto: if, and only if, the set E is in the class of measurable sets.
57:17:320Paolo Guiotto: So this says that, at least for every measurable set, you have a corresponding trivial, measurable function. Because trivial, because this function has just two terms, 0, 1,
57:30:30Paolo Guiotto: It's very simple, okay?
57:33:210Paolo Guiotto: Now, let's see why.
57:37:150Paolo Guiotto: let's see, Y?
57:40:250Paolo Guiotto: Because, we have to check…
57:48:860Paolo Guiotto: If the set were, this is our F,
57:53:830Paolo Guiotto: F belongs to an interval I, belongs to the class F.
58:01:100Paolo Guiotto: Well, now let's think about…
58:02:980Paolo Guiotto: So this is the set of points. So here we imagine this function is, of course, as a function, the indicator is defined on the full space X with values, which are real values. Actually, it takes all the value of 0, 1.
58:22:320Paolo Guiotto: Now, since F takes only values 0, 1, what can be said about the set of points X in the domain, which is the set capital X, such that indicator
58:37:940Paolo Guiotto: of set E belongs to the set I.
58:42:870Paolo Guiotto: Now, this quantity is only CO1.
58:47:460Paolo Guiotto: So we have 2, actually 3L there, 4 out there. This I does not contain 0 and 1.
58:56:540Paolo Guiotto: So an interval i that is now 0, 1. So any interval that does not contain 0 and 1. So, what are the X for which I, this will take with regards to i?
59:08:560Paolo Guiotto: Since this add-only zero one, and here there is no one, this there is no X, so the surface A.
59:15:80Paolo Guiotto: So we have this event.
59:19:460Paolo Guiotto: The set where F belongs to I, is equal to… empty. If,
59:28:250Paolo Guiotto: Both 0 and 1 do not belong to I.
59:32:790Paolo Guiotto: Remember that I is in the set of values taken by F.
59:38:110Paolo Guiotto: The domain here is any abstract object, okay? X. So we don't have to think anything special. So here, we are looking to the set of points where F belongs to R.
59:51:370Paolo Guiotto: So if this set, does not contain, modes 0, F1, so then it's no point to expose F of X will be there.
00:00:760Paolo Guiotto: Now, assume that this iron base is 0, but not 1.
00:05:980Paolo Guiotto: So, let's say that second case, 0 belongs to I, and 1
00:11:780Paolo Guiotto: is not enough. So… You may wonder what is the set of finitex for which the indicator belongs to add.
00:21:810Paolo Guiotto: Since you take, consideration with the value 0, there will be all points of X for which the indicator is equal to 0.
00:32:390Paolo Guiotto: But there won't be one point of X for which indicator is equal to 1, because 1 is not denied.
00:39:400Paolo Guiotto: What are the points for which the indicator is equal to zero?
00:44:270Paolo Guiotto: Then the case of k equals to 0,
00:48:530Paolo Guiotto: If and only if X is not in set G, so it's the complementary of E.
00:54:550Paolo Guiotto: So this means that, in this case, we get the complementary of E.
01:00:640Paolo Guiotto: Okay? This is the set of points for which the indicator gives value zero.
01:05:710Paolo Guiotto: Third case, we have that 0 is not in I, but 1 is in I.
01:11:980Paolo Guiotto: So this time, there will be, in the set F belongs to I, there will be all points for which
01:19:620Paolo Guiotto: The indicator, which is our F, gives value equal to 1.
01:24:690Paolo Guiotto: And that's the set E.
01:28:140Paolo Guiotto: And last, we have both 01 belong to I.
01:33:560Paolo Guiotto: Now, since i contains both 0, 1, which are all the possible values taken by our function f, the indicator.
01:42:620Paolo Guiotto: And now, it is clear that what is the set where the function, indicator, is 0 or 1? Everything.
01:50:450Paolo Guiotto: So, in this last case, we got the full space X.
01:54:280Paolo Guiotto: So… In, we see data.
01:58:220Paolo Guiotto: Whatever is the hit of our i, we have only 4 possibilities.
02:02:440Paolo Guiotto: The set where F belongs to I is either empty, E complementary, E, or X.
02:10:270Paolo Guiotto: Now, this belongs to the family F, for every i interval, Whatever the guy.
02:19:710Paolo Guiotto: Interval.
02:21:180Paolo Guiotto: If, and only if, these four sets belongs to the family.
02:26:940Paolo Guiotto: So, empty.
02:28:700Paolo Guiotto: E complementary. E. Index belongs to the family.
02:34:810Paolo Guiotto: But, the family is a sigma algebra, so it contains always, empty, and everything.
02:41:30Paolo Guiotto: So these two guys, we know that they, whatever it is the case, they belong to F.
02:48:110Paolo Guiotto: So, we see that what must happen is that both E and D complementary belong to F. But F is a sigma algebra. If E belongs to F, also its complementary belongs to F.
03:05:610Paolo Guiotto: And vice versa, if E complementary belongs to F, also complementary of E complementary, so E complementary. Complementary is E, belongs to F. So it means that, actually, this is equivalent to say that E must be in F.
03:21:740Paolo Guiotto: And that's exactly the conclusion. So we say that the function, the indicator, is measurable if and if the sector is in the family F.
03:34:790Paolo Guiotto: Now, of course, this is, this type of function is apparently far, very far, from…
03:42:940Paolo Guiotto: the kind of functions we may think that they may assume a continuum of values, but as you will see, they play an important role here. Now, we can extend a bit this example.
03:59:460Paolo Guiotto: Let's introduce the definition.
04:03:320Paolo Guiotto: We call… simple function.
04:16:250Paolo Guiotto: a function.
04:20:790Paolo Guiotto: S.
04:22:120Paolo Guiotto: defined on X, Read valued.
04:25:830Paolo Guiotto: Well, it could be even a complex value, we could also consider, but here we stay for a moment on a real value function.
04:34:740Paolo Guiotto: That… Bakes, sir.
04:43:330Paolo Guiotto: Only.
04:46:340Paolo Guiotto: Okay, fine, it… Number… of values.
04:59:250Paolo Guiotto: So this means that, if you want, we can, we can say equivalently this in many ways. For example, we can say that S of X, so the image.
05:11:460Paolo Guiotto: through the function S of the full space X is a set made by a finite number… a finite number of numbers. So, let's say C1, C2, etc.
05:25:320Paolo Guiotto: CN.
05:31:560Paolo Guiotto: Now, you can easily verify that a function of this type
05:39:130Paolo Guiotto: let's say it's a little, very little proposition, but I ask you to do the pro first.
05:46:990Paolo Guiotto: It's not particularly different from what we have done here. As you can see, the indicator is an example of a function of this type. It takes just two values, 0, 1, okay?
05:58:180Paolo Guiotto: So here we are a little bit extending this class by taking functions that assume any finite number of values, okay? Of course, C1 is different from C2, etc. here. So S is measurable.
06:17:450Paolo Guiotto: If, and only if, huh?
06:20:240Paolo Guiotto: Well, what is the extension of this condition that we have seen here? We have seen that the indicator is measurable.
06:31:160Paolo Guiotto: Now, how can we review this? Now, the set E is exactly the set where the function is 1.
06:40:480Paolo Guiotto: And its complementary is the set to have a function 0.
06:44:420Paolo Guiotto: These two sets are the sets where the function takes the two values.
06:49:760Paolo Guiotto: The idea is that the function is measurable equivalently, if and only if the second where the function is 1 is measurable, that's it, and the set where the function takes the other value, 0, which is equivalentally, is measurable.
07:04:970Paolo Guiotto: Now, this comes, in general, if and only if each of the sets were S is equal to CJ,
07:15:910Paolo Guiotto: for every J equal.
07:18:230Paolo Guiotto: One.
07:20:80Paolo Guiotto: to N. These sets are measurable for every J.
07:27:840Paolo Guiotto: Be careful, because, it is not true that, if you say.
07:34:190Paolo Guiotto: The function f is measurable if and if the set where f is equal to constant is measurable for every constant that supports. We will maybe return on this. But for discrete functions like this, this is true.
07:47:620Paolo Guiotto: Now, in particular, it is convenient to keep in mind this factor.
07:54:50Paolo Guiotto: Notice that, if you call EJ the set where S is equal to CJ,
08:04:560Paolo Guiotto: Since our function takes only values C1, CN,
08:10:80Paolo Guiotto: Now, this is a subset of X, okay? Because this is the set of X to be…
08:18:460Paolo Guiotto: formally precise. It is the set of little x that belongs to the space capital X, such that S…
08:28:279Paolo Guiotto: of X is equal to the constant CJ.
08:34:520Paolo Guiotto: Okay.
08:36:10Paolo Guiotto: No? So this is a subset of S, and it is a measurable subset if the function S is measurable.
08:45:00Paolo Guiotto: Now, you can easily see that the function S
08:48:510Paolo Guiotto: can be written also in this form. It is the sum…
08:54:229Paolo Guiotto: for J going from 1 to capital N of CJ times the indicator of the set EJ
09:05:260Paolo Guiotto: This factor, and this is the standard way to represent these kind of functions, these simple functions. So sometimes you see the definition. Actually, the simple function is it has this form.
09:16:880Paolo Guiotto: Okay.
09:18:130Paolo Guiotto: Now, we say that that is simple if it takes on a finite number of times.
09:22:590Paolo Guiotto: Because this song is actually a sort of fake song, huh? Since the EJ, you can see, he's John Luther, because…
09:30:340Paolo Guiotto: EJ is the cell when S is equal to CJ. So, since the CJ are different, you cannot have a quant that is in E1, that S is equal to C1, and it is also on 2.
09:43:109Paolo Guiotto: where S is equal to 2. Now, the setup is geometer. It means that only one of these terms is different from 0, and that for that term, the indicator is 1, and the value is 0.
09:56:790Paolo Guiotto: exactly when X is in detail, this sum is about to say, so they are the same. And this is a standard way to represent P.
10:06:170Paolo Guiotto: Simple functions.
10:12:770Paolo Guiotto: Okay. So, in some way, you see that these simple functions are linear combinations of indicators.
10:23:550Paolo Guiotto: So, in fact, we are not leaving too much the class of indicators, even if we are building more complicated functions. Of course, we may expect something about functions
10:35:600Paolo Guiotto: would take a continuum of values. That's a bit more complicated. So, let's see some other characterization. Maybe we could see the proof, because it is basically… the proof are based on working with the definition. So, we have this proposition.
10:56:180Paolo Guiotto: Something which is useful to know, because sometimes it is not… the best,
11:03:850Paolo Guiotto: to check, with this condition if the function is measurable or not, okay? So sometimes we…
11:11:790Paolo Guiotto: may have some advantage by using some equivalent characterization. So, the following…
11:23:920Paolo Guiotto: properties.
11:29:600Paolo Guiotto: are equivalent.
11:33:260Paolo Guiotto: So, number one, F is measurable.
11:40:90Paolo Guiotto: Sometimes you read F measurable because the measurability is not a property that depends on the function alone.
11:49:450Paolo Guiotto: It is always depending on what is the sigma algebra you have on the space.
11:57:190Paolo Guiotto: Now, in analysis, normally, you work with the Lebec measure, there is a unique sigma, which is the Lebec class, and more or less, there is no question about what is the environment.
12:09:30Paolo Guiotto: In probability, often, you work with different sigma algebras, because, for example, sigma algebras are used in probability, sigma algebra are families of events to which you assign the probability.
12:22:790Paolo Guiotto: Different sigma algebra are used to do… to describe different available informations.
12:30:450Paolo Guiotto: So I have a big sigma algebra, means there are lots of events to which I can assign a measure. A smaller sigma algebra, maybe, is a sigma algebra made by a smaller number of events. It means less information, because, you know, only these things may happen, I don't know the other, okay? So…
12:47:890Paolo Guiotto: This is to say that in probability, working with different sigma algebras is normal. And therefore, it makes sense to specify, you are measurable with respect to this, you are measurable with respect to that.
13:01:370Paolo Guiotto: Okay, sometimes, so you see that here, instead of writing is measurable, is F measurable, to say it is measurable with respect to the family F.
13:11:780Paolo Guiotto: So what has been… what it must be clear is that measurability is not a property of the function alone. It depends also on a sigma algebra, okay? It comes with the sigma algebra, not just because the function is this or that.
13:27:970Paolo Guiotto: Number two…
13:30:50Paolo Guiotto: instead of checking F belongs to interval is measurable, you could check, for example, a set like F greater or equal than A,
13:40:680Paolo Guiotto: is measurable for every A real.
13:45:410Paolo Guiotto: Or number 3, which is, as you may imagine, a slight variation of this, set like F strictly greater than A is in the class for every area.
13:57:50Paolo Guiotto: So these three are equivalent.
14:00:410Paolo Guiotto: So, let's see what… maybe we will not do all the proof, because some… something is a repetition here. Well, let's see what is trivial. For example, the first implies the second is trivial.
14:19:440Paolo Guiotto: Why?
14:20:990Paolo Guiotto: Because… so, suppose that you know that the function is F measurable, that's the assumption. Hypothesis, F is measurable.
14:33:70Paolo Guiotto: thesis, we have to prove that the set where F is larger or equal than A belongs to the family F.
14:40:980Paolo Guiotto: But that set is a set of type. This means that F belongs to the interval from A to plus infinity, you see?
14:53:950Paolo Guiotto: That's an interface, the same thing, no? F of X looks way less affinity if f of x is greater than A, so they are the same.
15:04:470Paolo Guiotto: And this is an interval, so you know that by definition, by definition.
15:10:800Paolo Guiotto: This means that whenever you have F belongs to I, this is a measurable set for every i interval.
15:20:440Paolo Guiotto: So that's what you know. You know that F belongs to I, is measurable, whatever is i. And therefore, in particular, this is an interval I, and you can conclude that this belongs to F. So that's why it's trivial.
15:36:230Paolo Guiotto: less trivial.
15:38:620Paolo Guiotto: is the vice versa. Let's see how much is, complicated.
15:44:850Paolo Guiotto: The vice versa. So now we know
15:48:20Paolo Guiotto: the hypothesis is that the set where F is larger or equal than A belongs to class F for every A,
15:58:460Paolo Guiotto: And the thesis is that we want to prove that F belongs to I, no matter who is the interval I, is in the class F. Let's see…
16:09:540Paolo Guiotto: what has… what we have, to do to check this, no? So the thesis is, prove that F…
16:19:440Paolo Guiotto: Belongs to I.
16:21:460Paolo Guiotto: is in the class F for every I interval.
16:25:930Paolo Guiotto: Now, we may say that, because of the previous argument, we have a special case of the thesis, no? So we know that the thesis is true when the interval i is a half line A plus infinity.
16:40:30Paolo Guiotto: So… Mmm, this is true.
16:45:400Paolo Guiotto: by… hypothesis.
16:49:100Paolo Guiotto: when… I is a closed off line, something like A2 plus U2.
17:00:590Paolo Guiotto: Of course, I interval means that there are many other cases, no?
17:06:40Paolo Guiotto: So, I, for example, could be an open offline. So, let's see what happens here. What if…
17:16:820Paolo Guiotto: I is not the closed line with A, but the open. It's just the detail, apparently. But this
17:25:690Paolo Guiotto: Makes, this case a little bit more trivial.
17:29:600Paolo Guiotto: Because, what we know is that we have… that these type of sets are measurable.
17:35:420Paolo Guiotto: And we want the set where F belongs to I, in this case, becomes the set where F belongs to the interval A to plus infinity, so it means the value of F is strictly greater than A.
17:51:630Paolo Guiotto: It seems the same, but there is a little difference.
17:55:680Paolo Guiotto: And that little difference makes this factor not immediately clear.
18:03:600Paolo Guiotto: So, how I can deduce from this…
18:08:180Paolo Guiotto: Now, the best thing to do here is to change the letters somewhere, because you may get confused by the use of the same A. So let's use here a different letter, so let's call this guy, for example, B.
18:23:950Paolo Guiotto: I know that the set where F is larger or equal than B is measurable. I want the set where F is strictly above A measurable.
18:34:460Paolo Guiotto: So how can we do that?
18:37:490Paolo Guiotto: Now, let's think about…
18:40:430Paolo Guiotto: the set where F must be strictly above, then, A. Fix A in the real line, you want that values F of X
18:50:950Paolo Guiotto: are here, strictly a door.
18:54:510Paolo Guiotto: well, you understand you're looking at this figure because you are strictly above, that if you are strictly above the value A, you will be…
19:05:510Paolo Guiotto: Greater or equal than some value.
19:09:110Paolo Guiotto: at right of A, so something like A plus epsilon, with epsilon sufficiently small.
19:18:280Paolo Guiotto: In fact, I could say that, for example, take as…
19:25:190Paolo Guiotto: Normally, when we take an epsilon small, we can take quantities, concrete quantities, like 1 over n, no? So, there will be an n for which my F of X will be greater than 1 over n.
19:38:620Paolo Guiotto: So I can say, let's write this idea formally.
19:42:50Paolo Guiotto: my F of X is greater than A,
19:47:130Paolo Guiotto: I can say that for the moment there is a one-way arrow, then…
19:54:330Paolo Guiotto: Since, as you see in the figure, taking N large enough to exist
20:01:150Paolo Guiotto: And then, such that my f of x will be larger or equal than a plus 1 over n.
20:10:610Paolo Guiotto: Of course, not for every n, because I don't know where is this F of X, it's just that the Y of F of A, no? So maybe it's very close to F away.
20:23:160Paolo Guiotto: And these are A plus 1, I don't know, here there is no scale, but maybe A plus 1 is here, A plus 1 half is here, A plus 1 term is here. What you see is that sooner or later, this number would be so small that A plus 1 over N would be at left of f of x.
20:41:200Paolo Guiotto: So you can say that if f of x is greater than A, it's 3 equity, then if it exists at least 1 n, there are completely many equals n larger than this one parameters, such that f of x is going to depend on A plus 1 over n.
20:56:270Paolo Guiotto: Actually, this is a significant belief, because think about it, if this happens.
21:01:390Paolo Guiotto: It means that N of X is greater or equal than this, which is greater than A, huh?
21:07:860Paolo Guiotto: This number is automatically greater than A.
21:12:470Paolo Guiotto: So, this means that this is if and only if.
21:16:350Paolo Guiotto: And this means that we have a characterization for the set where f of x is larger than that way, because the left-hand side says that
21:28:70Paolo Guiotto: X belongs to the set where F is larger than A.
21:33:160Paolo Guiotto: Now, this is equivalent to say that X belongs to what?
21:37:580Paolo Guiotto: Now, this is a bit difficult, but you must get used to do this, because even when we, we do probabilistic assessment, when we assess the probability, we need to write the intervention in some way. Normally, we know, but in the set, then we have to transform into set operations.
21:57:430Paolo Guiotto: So now… What does it mean, this? So, this is saying that X belongs to the center where
22:05:630Paolo Guiotto: F is larger or equal A plus 1 over N. So this is saying, so let's say, let's do one…
22:16:40Paolo Guiotto: There exists such that X belongs to the set where F is…
22:23:550Paolo Guiotto: Larger or equal A plus 1 over N.
22:28:340Paolo Guiotto: Okay?
22:30:640Paolo Guiotto: But if there exists an end such that you are there, it means that X belongs to which set?
22:41:440Paolo Guiotto: I take these sets, they are infinitely many, because for every N, I have Is that the…
22:49:120Paolo Guiotto: And what is the equivalent? It's what I'm making right here. Exactly, the union, no? Because if you put the union together, or in this period, this is a… you are in one of them, you are in the union. That's true.
23:04:940Paolo Guiotto: But, if you are in unit, please, you are at least in one of them.
23:09:50Paolo Guiotto: And you get this one. So the equivalence is exactly this. This is the union
23:14:850Paolo Guiotto: of a N of this.
23:17:400Paolo Guiotto: So this means that we are… we have been able to write this.
23:21:800Paolo Guiotto: our set F greater than A,
23:25:20Paolo Guiotto: Has been written as the union of.