 To make this other notions bit more formal for the proof, we will just try to understand what we mean by continuity of probability. We just now understood what is continuity of function. But probabilities, we say probabilities is also a function, but probability is function on what? Probability is defined on what? Events. Probability is a function from events to, yeah let us say 0, 1 or real number if you want to make it more general. But now when we are talking about probability, it is defined on sets, not points, right? Now when you are going to, so when we define continuity in at least in this one dimension, we defined on points and our definition included that right continuity, left continuity and all, right? To define right continuity and left continuity, we had a sequence of points. Like here when we said it is continuous point, that means we took right limit sequence of points converging to x2, sequence of points converging to x2 from left and right and then verified they coincide. But now if we want to define continuity of probability, we need to have a similar notion like of this convergence of points, but defined on sets, right? Because probability itself is defined on sets. Now how we are going to have this? What is the notion of convergence in sets? So we are going to, so let us say we have this sequence of sets. So when we looked at sequence of points, I looked at points which are, so let us say I want here, I want a sequence xn converging to x in this case. I can get this sequence xn converging to x in multiple ways, right? One possibility is this xn convergence to x in this fashion. What I mean by this or sorry xn is increasing sequence and it converges to x, right? That means I had a sequence which is approaching this point x2 from left, right? So all the sequence here has increasing and eventually they are converging to x2. Other possibility is I could approach the same point x2 from right and that sequence I can write it as like this, so here I am coming. So when I am defining these limits, I am looking in this case the right and left limits as sequence which are monotonous here, right? Like I am going to get a monotonously increasing sequence when I look for left sequence and I am going to get monotonously decreasing sequence when I look at the right sequence that is converging to the point x. So let us say we also look at the sequence of sets and we are going to define the limit on this like monotonically increasing sets. When I say monotonically increasing sets, that means they become larger and larger. B1 is contained in B2, B2 contained in like this. Now in this case, what is the notion of limit? So this is nothing but I have this sequence of BNs. Yeah, so we can consider that. It could be included. So that is what I said, right? When it is included, I am not going to write. When it is not included, then I am going to write it like this, okay? If I just write like this, that means B2 could be same as B1. Now suppose if you have a sequence like this, what do you expect the limit to be? Why sample space? Yeah, the biggest one. How does the biggest one look like? Yeah, it could be possibly union of all this, right? So I could write limit of n tends to infinity bj as union of bj, j equals to 1 to infinity. So bj's are increasing sets, right? Anything at j tends to infinity, whatever the set we have here, that should set, should be as much as you can verify this like. So how you are going to verify? Like if you are going to let j go to infinity, whatever the set, if in that limit, if a omega point is there, is that omega point also belongs to here. And now if you take a omega point in this set, will that be also belonging to this sequence as j tends to infinity? So because of that, we can verify that. I mean, this is just like definition. We will just take it like a definition and this makes sense. Now, we have a equivalent notions of xn converging to point here, but in the set, in the onsets, okay? So similarly, if we have b1, so how you like this to be defined as? What should be the natural vinyl set? It should be an intersection, right? So this is like a decreasing sequence, right? Whatever the value you have eventually, that should be at the intersection of all the points. So this thing here is an analog of xn converging monotonically, xn is a monotonical sequence converging to x2 and this is an analog of x2 is a monotonically decreasing sequence converging to x. So now let us define, so okay, before I make, so if this function f is continuous at point x2, okay, let us take this. I have a sequence xn which converges to, instead of x, let me call this y. I have a sequence, let us call this as some point y. So what does this mean? Limit as xn n tends to infinity equals to y. If f is continuous at y, is this true? If the function f is continuous at y, do you expect this to hold? This is true, right? Irrespective of how this sequence xn is converging to y, either from the left or right, we do not care or it is a mix of both left and right. So any sequence xn that is converging to y should satisfy this property. And if this holds, then we say that my function f is continuous at y. So now let us apply similar notion here. So with this, we will show that this is, let us write it as lemma. So this is our claim. Or maybe the better way to write is let p be a probability function, let bn be a sequence. So what we are saying is let p be probability function and you are given a bn sequence. And now if this bn sequence whatever you are given is monotonically increasing, then this p function will satisfy this equality which says that if you are going to take this probability on this sequence pb and compute their limit, that limit is going to be nothing but probability on the union of the sets. It is just clear. And similarly, if this b sequence is like a decreasing sequence, then if you are going to apply this probability on the sequence and take the limit, this is nothing but the limit is nothing but the probability on the intersection of these events. So in a way what we have said here is, so the other way of writing this guy here is limit j tends to infinity p of j is equals to probability limit of j tends to infinity of bj. So what you have basically done is I have just replaced this definition. This union of j 1 to infinity is nothing but what? Limit as j tends to infinity of bj. This is our definition. Now what we are just saying is if you want to take a limit of the sequence of the probabilities, this is nothing but take the probability of limit of bj. That means we have basically interchanged this probability function and limit. So now first we took the limit and then we are applying, taking the probability, taking the limit. It is saying that this is going to be same as first take the limit and then apply the probability on it. So we are just, whenever you have this limit, the probability function is such that it allows us to interchange between limit and p. That is exactly what happened here also. So what was y here? Here y was limit was n tends to infinity of xn. Now, so this f of y was nothing but limit of f of xn as n tends to infinity or maybe I should have written this one in a slightly different way. So, if f is a function continuous at y, we know that limit as n tends to infinity of f of x equals to y, but this is nothing, what is y? y is nothing but limit is infinity of y. So this is the definition of y. So here if f is a continuous function at a point y, at that point for any sequence that converges to y, I am able to interchange this limit and f in this fashion. So this was the definition or this was the implication of my function f being continuous at y. And this is what I am also saying similar properties for my p here. p also I am sure we have what, I have not assured this, but what I have shown, what I have, the way I have written or claimed is, it shows that I can interchange the limit. In this way, I can, can I interpret this function p is a continuous function, the analogy with this guy here. So whenever f was continuous, I am able to interchange the sequence and the limit or the limit and the function. So here I am also doing this function. So because of this, this property is going to be called as continuity of probability. Let us quickly show this. This will, I mean all of you are with me like, this makes sense to claim that p is a continuous, probability is a continuous function in some sense because of the analogy we draw from what is the property of a continuous function. So these are finer aspects we need to see which we are going to use later as I said to prove that part. So now let us try to argue why this should be true. So I have this increasing sets of sequences. Let me call this as b1, this is b2 and b3 and like that we have. So now I am going to define this to be d1 and the difference here to be d2 whatever that is there between b1 and b2 and similarly what is there between b2 and b3, let me call that as d3. I can keep on doing this for everybody, right? That means, so what I have basically done is I have taken di to be bi and I am going to define it as null set. Is this fine? Can I define my sets like this iteratively? So this is, I have another sequence di and is this true that union of di is equals to union of bi? Yes, no because they are basically capturing the same elements, right, together. I mean you just take b1, b2, they are union or just take their difference whatever the increment we are going to get from b1 to b2 and b2 to b3, you just take their union you should get the same thing. But what is the difference between these two? While the sequence bi's are nested, these di's are which will be exclusive because of the way we have defined it. So is that clear? I can write this as union of di same as union of bi, what I want to show? I want to show this, right? Let me take this limit as j tends to be p of bj. I am now going to write it as pj is equals to p of di i 1 to j. Is it correct that probability of pi is equals nothing but the sum of pdi's from i equals to 1 to j, right? Because bj can be represented as union of di's and there it is joined and from that property of di's I know that the finite additivity property satisfy, right? The probability of union of di is nothing but summation of probability of di's, that is what I have exactly applied. So this was one of the properties or the axioms that we assumed p should satisfy, okay? So now by definition this is nothing but i equals to 1 to, so I have just like this is a, I am just letting j go to infinity, right? I am just letting, just taking this upper limit to be infinity that is was the definition of di and now I know that this di's are what? Disjoint, right? Now what property can I exploit here? If I want to write it as just one probability here instead of the summation of the probabilities, probability of union of di, right? This is just the meaning of sum of probabilities of disjoint sets. Now we already know that this union is nothing but union of bi's because they are covering the same set of element. So this should be equal to then probability that union of di i equals to 1 to infinity and this is what we wanted to show here, right? So if you want to show this on this set, on the decreasing sets, what is the changes you can possibly think of? How you are going to, so you have to now define your di's directly, right? How you are going to do that? Largest one at infinity, you do not know. So think about this like how you are going to do, so just do that as an exercise, okay? You should be thinking along exactly the same line, okay fine. So ultimately from this, so all this detour is to just to convince ourselves that if there is a probability function, I am able to interchange limit and probabilities. Just like from the whole detour, just we are going to just take it that like, okay fine. Whenever I have probability function, I have a sequence of probabilities, so sets that are monotonically increased, I should be able to increase, I should be interchange probability and the limit in this fashion. That is the only thing I wanted to use from this part, okay. Now let us go back here. So do you think this is obvious? So let us come back to the properties of CDF. This is a cumulative density function, right? So accumulation means you should keep on accumulating and what we are accumulating, we are accumulating probabilities which are non-negative quantities. So it should be, this function should be increasing, so how to show this formally? So see this, I have said if and only if, right? You guys understand what I mean by if and only if? So what does that mean? So what we are saying is suppose if CDF, if I am saying CDF, then this is CDF for already some random variable, right? Then it should be satisfying all these properties. And now if F is some function which satisfies this, then it should correspond to CDF of some random variable. What is that random variable? We do not know right now. It depends on what is the F we are talking about. So in this, we are only going to show only if part. That is if F is the CDF of some random variable, it is going to satisfy these properties, okay. In the other direction we skip. Okay, how we are going to show F is non-decreasing? If you are going to take X greater than Y, then we need to show that F of X is going to be greater than or equals to F of Y, right? This is the meaning of that F is monotonically increasing. So let us take the case only X greater strictly greater than Y because if X and Y are same, nothing to show, right? So let us take X is strictly greater than Y and then try to show that F of X is strictly greater than or equals to Y. How we are going to show this? So by definition, F of X is going to be less than or equals to X and then this is the case if Y is strictly less than 1, it should be the case that X is less than or equals to Y plus X is, can I split this probability like this? So what I have basically done is I have taken X is less than or equals to, so we have, the way I have done, first comes Y, then comes X. So probability that X takes value less than or equals to this is same as probability that X takes value less than Y and probability that it takes value in the, between X and Y. Now is it obvious that this should be equals to F of Y? Why is that? So we know that this guy is going to be greater than or equals to 0 and this is nothing but F of Y. So now let us try to show the second part. Now to show the second part, let us take a sequence BN, X is less than or equals to N. So before I do this, when I say only if part, right, I am going to say that F be a CDF for some X and then I am doing this. I am already assuming that F is CDF of some random variable and I am doing this. So let us take for that random variable, let us define a sequence like this. Now if I look at this sequence BN, is it monotonically increasing? Yes or no? So that means that X I am increasing N, right? So it should include more and more elements from my sample space. So that is why this is a monotonically increasing function. Now what is F of N? It is nothing but probability of B of N, right? By definition F of N means probability that X is less than or equals to N. That is exactly BN. Now if I let N go to infinity and now at this point I want to exploit the fact that my P is probability function P satisfies continuity property. So if that is the case, how can I write it? Now what is this union? Now just so what is BN? BN is X is less than or equals to N and now I am allowing this N to go to infinity. So what is this quantity is going to be? Sample space, right? So this is P of omega equals to and I know that is going to be 1. Am I done with this part of the proof? Yes, from this. So does this, whatever I have just shown here, does this concludes the proof of this part? Yes. But I have done it only for N integers, right? But whereas this X can be any real number. After that what does not matter? Yeah, so what cumulative value? Why is that like okay? So what you are saying? Fine. According to this definition, so FNs are now real numbers, right? Yeah, so we have this definition. According to this, for all epsilon greater than 0, there exists some N epsilon such that for all N greater than or equals to N epsilon I have F of N greater than or equals to 1 minus epsilon, right? This is the definition of the limit. You take N sufficiently large then in epsilon F of N is going to be less than N. Now instead of this N, now you look for all the points beyond this N, any point beyond this N and my function is such that it is non-decreasing, right? It is increasing. So all the points above this N should be also be greater than 1 by epsilon, right? So because of that if I am going to took all the points which are greater than N and also if I look at all the points between N and N plus 1 like that, we can convince yourself that this is indeed true even if I replace N by X, any real number which is instead of the X, we can say that for all X greater than or epsilon this is going to happen and because of that this is true that I can assume X equals to infinity limit of F of X equals to 1. So you have to convince yourself that yes I can replace because of this monotonicity property of my function F even though I have shown it only for the integer value, I can replace it by the continuous real numbers, okay? And then the very definition of the limit says that this is true and this is like in a similar fashion you can show this. All you need to do is replace N by minus N, okay? So I wanted X go to plus infinity, right? Now I want to go X to minus infinity. So replace BN by B minus N that means X less than or equals to minus N. So in this case what about my sequence will be? It will be decreasing sequence now instead of the increasing sequence. But even for the decreasing function your probability definition that the limit and probability you can still interchange, right? And then you do the same thing, but now you are coming from the in the negative direction. That is why this should be true. The last property. So what I am saying here is F is right continuous. When I say I did not say F is right continuous at some point, right? I just said F is right continuous. That means I need to show at any point X my function F is right continuous, okay? So take an arbitrary point X R and we have to show that for whatever the arbitrary point X you are given my function F is right continuous. So now I am going to define a sequence FN equals to X. So this X is the same point whatever you have taken and now for any N I have defined a sequence like this. Now is this sequence, this sets now A N is a set, right? Is this increasing set or decreasing set? It is going to be decreasing, right? As you increase then this is going to be shrinking X plus 1 plus N. So now again apply our standard trick of using continuity of probability here. So this is nothing but limit of N tends to infinity of probability of A N, right? I just use that like this quantity here F of X 1 plus 1 1 by N is nothing but probability of A N by our definition of A N. And now because this A N is a monotonically decreasing function and by the continuity property of P how can I write? This is nothing but probability of intersection of, okay? Now go back to the definition of A N. A Ns are looking like this. What is going to be the intersection of all these A Ns? X or 2 X are there. It is going to be like if I am going to let N go to infinity here the limit this will be simply equals to X, right? Because this is nothing but this is a sequence which is approaching from right to X, okay? And now that is why we can say that this is nothing but this is the limit of this. So now what I have basically done is and this is nothing but F of X by definition, right? What I have basically done is I have taken a sequence here which is approaching X from the right, right? So this sequence as N goes from 1, 2, 3, 4 up to infinity this is approaching from right the quantity X here. So I have constructed a right sequence which is approaching X from the right and for that we have shown that this is true. But to show right continuity what we need to show take it any arbitrary right sequence that is converging from the right and show that this holds, right? So now this by this itself my proof that F is right continuous is not complete, right? Because I have what I have demonstrated through this is there exists a particular sequence that is converging to X from the right and where this holds. But what about the arbitrary sequence that is converging? Do you think using this argument you can extend this same analysis to any sequence that is converging to X from the right using which property? Continuity I have already used. So do you think we can use again the monotonicity property to make this argument work for any sequence? So check that this one which I have proved it for a particular sequence the same argument can be used to argue that for any right continuous sequence this is true. How is that true? Like this is going to be the same argument like this, right? Here I showed you for integer value. So here in this proof we show that we can extend the argument for any X using the argument for that holds only for the integer values, right? It is also similar case here. Now we have an instead of n we have just 1 by n here. We have an argument that construct a sequence using this integer valued sequence. I mean not integer valued but a sequence. But now you can show that using this you can come up with this argument extends to any sequence that is right converging to X. So just see that like 1 by n is converging to 0, right? So at some point it should be falling arbitrarily close to 0. If it is a right continuous sequence so any right side continuous sequence also like let us say this is any right continuous any sequence that is converging to X from the right. So if that is the case this Xn should be also going to 0. It is just let this 1 by n is going to 0. So whatever the values in between that you can if it has any other sequence it should be falling between one of those n, 1 by n and 1 by n plus 1. And then you can say that this is true for any arbitrary right sequence that is converging to X from the right. So I hope that you will convince yourself that this is going to work for any arbitrary sequence that is converging to X from the right. So then we are done with all these 3 points. So before we leave can you find so we just define some things and try to prove the properties. So do you think where the CDF is going to be useful? Do you think CDF is going to be useful at all or just like fine? You just defined it and do whatever you like to do with it. So if the CDF function is already available we already know that probability right. So if you have this pre-computed CDF function we already noted what is that suppose the way the temperature varies you have a distribution on that and you know already the CDF. So if you want to know what is the probability that my temperature today in Mumbai will be less than let us say 30 degrees just look at the CDF and you already have that. We just show that like how that CDF what should be the properties of the CDF and this is going to be help lot further as we will see.