 So, last class we were dealing with the problem of trying to define a uniform measure on the 0 1 interval. So, we say 0 1 interval is of course, an uncountable set is an uncountable sample space. And we were trying to see if a translationally invariant measure uniform measure could be defined on the 0 1 interval. So, it so happened that we stated an impossibility result. So, we said that if you want to define this uniform measure on all subsets of the 0 1 interval like the assign measures for all uniform or for all subsets of this 0 1 interval this uniform measure is not possible. So, that is the impossibility result we saw. So, what the impossibility result says is that even something as simple as intuitively simple as uniform measure cannot be defined on 2 power omega when omega is uncountable like 0 1 interval. So, we have to so what this says is that if you want to define a uniform measure on 0 1 you have to make a compromise and work with a smaller sigma algebra this is what the conclusion was the end of the last class. So, now the question the question that arises it what should be the smaller sigma algebra. So, if you take 0 1 2 power omega is this 2 richer class of sets to assign probabilities to assign this measure to. So, we have to settle for a smaller sigma algebra and what should the smaller sigma algebra be. So, you have the 0 1 interval. So, you want to include as your interesting sets sub intervals of this intervals. So, you want all intervals of the form a b or close j close b open here close here you want those sets to be considered interesting sets. You want those intervals sub intervals of 0 1 to be in your sigma algebra, but in order to see the sub interval the set of all the collection of all sub intervals of 0 1 do not make a sigma algebra. So, if I take an interval like a b the complement is not an interval for example. So, in order to make this a sigma algebra it has to be closed under complementation and countable unions countable intersection. So, that is what we are building towards where we want a sigma algebra which contains all the intervals and therefore, contains all complement of intervals and countable unions of intervals and that sigma algebra we are building up towards is called Borel sigma algebra. Now, but we have to do this formally. So, this is the topic now. So, this is called generating generated sigma algebras. So, what you have is the following let us see this scripted C b an arbitrary collection of subsets of omega. So, you are given some arbitrary collection of subsets of omega the scripted C. So, this C you should intuitively think of as being your interesting subsets the subsets you want to keep in your smaller sigma algebra. So, what you are looking for is you have decided that this collection C of subsets is interesting to you for example, the 0 1 case we said it should be intervals sub intervals of 0 1. So, the C is that collection that is that you think is interesting to you now you want to generate a sigma algebra which contains all the C all the elements of the C. But the C need not be a sigma algebra with C is just a collection of interesting subsets, but you have to make a sigma algebra from it that you have to generate a sigma algebra from it. And what you want to do is you want to generate the smallest sigma algebra with the simplest sigma algebra in some sense that contains all elements of C that is your task is it clear what we are trying to do. So, in the case of intervals sub intervals of 0 1 you want to look for the smallest sigma algebra that contains all intervals. So, let us keep it more abstract at this point let us see let omega be some sample space and C is some arbitrary collection of subsets and you want to define the smallest sigma algebra that contains all subsets in C. So, there is a theorem that I am going to state it is a very simple theorem it says that there exists a unique sigma algebra which is the smallest sigma algebra that contains C. This is actually a fairly straight forward result. So, I am saying that there exists a notion of the smallest sigma algebra that contains all elements of C. So, what do you mean by smallest sigma algebra. So, you understand see you understand the C is some collection of subsets I am looking for some sigma algebra that contains all the subsets. But, I want the smallest such sigma algebra and I am saying such a sigma algebra exists right. . . . . . . . . . . . . . . . . . See C is scripted you see it is not a set it is not a subset of omega it is a collection of subsets of omega. . . . . Now, I am keeping it abstract, but in the case that omega equal to 0 1 you can think of C as being the collection of all intervals of the firm A B right. If you want a concrete thing to think about right. So, I am saying that there exists a smallest sigma algebra containing the C. Now, what do I mean by smallest sigma algebra? . . . . . . . . . . . . . . . . . . . . . . . .If you find any sigma algebra that contains C then sigma C is contained in H that is what I mean. So, if you find any sigma algebra H which contains my collection C then what I refer to? So, what this sigma C is the smallest sigma algebra I am referring to that guy is contained in H. In that sense it is the smallest sigma algebra containing C. I still have to prove it of course right it is actually very simple proof. So, the statement clear and the sigma C is called. So, this is terminology again sigma C is called the sigma algebra generated by C. So, I have to prove the existence of the smallest sigma algebra and that essentially the smallest sigma algebra that contains all these elements of C is referred to as the sigma algebra generated by the collection C. . So, the proof of this theorem is actually constructive. So, what you do? So, you have given some collection C and you are looking for the smallest sigma algebra containing this collection C. So, what you do? So, the first thing you will do is go and find all sigma algebras that contains C. See on omega there are many sigma algebras one such sigma algebras 2 power omega, but there may be other sigma algebras that contains C. You go and find all such sigma algebras there may be an infinity of them there may be in fact an uncountable infinity of them right, but you go and find them all right. Let F i i belongs to i be the collection of all sigma algebras that contain you go and collect all sigma algebras that contains C. So, this collection of. So, this is the collection of sigma algebras sigma algebras are themselves collections of subsets right and this is the collection of sigma algebras all of which contain C which is the collection of interesting subsets right. Now, is this how many elements will this I mean. So, this collection of sigma algebras right can have infinitely many elements in fact you can even have uncountably many sigma algebras that contains C. Can it be empty why power set will be there yes yes yes. So, this collection certainly contains 2 power omega right this collection of sigma algebras that contains C definitely has 2 power omega in it there may be other sigma algebras they may not be other sigma algebras, but certainly this collection is not empty right. This collection contains 2 power omega and is thus non empty right. So, the claim is that. So, my claim is going to be the following. So, this sigma C that whose existence I want to prove as the smallest sigma algebra is in fact the intersection I belong to I this is scripted I of F I. So, what am I doing. So, you go and collect all sigma algebras that contain C and intersect all of them. Now, what do I have to prove I have to prove that this guy is in fact the smaller sigma algebras containing C. So, I have to prove that it is a sigma algebra first of all then I have to prove that it contains C. Then I have to prove that it is a smallest that sigma algebra right. So, the 3 things I have to prove right. So, in order to prove that this is a sigma algebra first of all you have a homework problem right. Homework problem says arbitrary intersections of sigma algebras are in fact sigma algebras. So, you intersect an arbitrary collection of sigma algebras you get a sigma algebra you take unions you do not get unions of sigma algebras do not lead to sigma algebras by intersections always lead to. So, this is. So, this guy is a sigma algebra very trivial that does this intersection contains C why because each of these F A's contains C right. So, this intersection contains C. So, what have we proven we have proven that sigma C is a sigma algebra that contains C. Now, we have to prove that this is the smallest that sigma algebra right. So, those 2 first 2 steps is very easy I am skipping I am not writing down all right. So, in order to prove. So, let H be sigma algebra such that C is in H. Now, I have to prove sigma C is contained in H correct how do I prove that. So, I am just saying that H is a sigma algebra that contains my collection C right. So, which means H must be one of these F A's right. So, F A is what this guy this collection of sigma algebra contains all sigma algebra that contains C. So, H must be one of the F A's right clearly H is one of the F A's right meaning that. So, in order to write if you want to write this more formally there belongs some I in this index set I for which H is equal to F I correct. So, then. So, this implies that sigma C is contained in H because sigma C is the intersection of all these F A's. But, one of these F A's is in fact H right. So, you are done any questions so far. So, as an existence theorem right. So, given any collection of subsets of omega you can find the smallest sigma algebra that contains this collection right. So, the smallest sigma algebra I want to point out that the smallest sigma algebra that contains this collection could be 2 power omega itself or could be some smaller sigma algebra right. But, it is definitely well defined right. So, it could so happen that the only collection only sigma algebra that contains C could be 2 power omega in which case bad luck right. In this case the smallest sigma algebra that contains you C is only 2 power omega right. But, it could be smaller there may be other sigma algebra that contain it right. So, the sigma this sigma C is well defined as the smallest sigma algebra that contains my collection C. No sorry C is some collection of subsets right. So, what is your question? My question is if all F A's must contain C for them to be sigma algebra. No that is not what I am saying F A's are those sigma algebras which contain C there may be other sigma algebras that do not contain C. No no no no no sigma the sigma algebras are defined over subsets of omega and C is some collection of subsets of omega correct. So, far. The second is the smallest we should show that there cannot be any subset containing C. In this case the smallest sigma algebra. So, by smallest sigma algebra that contains C I mean if there is some other sigma algebra that contains C that sigma algebra must be larger in this sense that is all. Any other questions? So far. So, now this is the theoretical construct behind this Borel sigma algebra. So, what we are going to do now is say omega equal to 0 1 interval and then C S as what? Sub intervals of 0 1 and then I am going to generate a sigma algebra using this construct right. So, I am going to talk of sigma algebra generated by intervals in fact. So, sigma algebra generated by intervals is your Borel sigma algebra. So, let me say this more properly just for the minor points. So, I will take for to be pedagogically sound. So, I will take open 0 closed 1 for reasons that will become clear later. I am not taking closed on both sides open on both sides and I am going to take C naught as sub intervals of omega sorry open sub intervals of omega. So, I am going to take open 0 closed 1 as my sample space not such a big deal I tell you why I am doing this. And I am going to take open sub intervals of this interval as my C naught C naught is my interesting collection. And I am looking for the sigma algebra the smallest sigma algebra that contains all my open intervals. Now, I am having only open intervals definition sigma C naught is called Borel sigma algebra 0 1 denoted by scripted B 0 1. Again this Borel sigma algebra is a collection of subsets right. So, I have to use it use script to denoted scripted later denoted. So, I denoted by scripted B of 0 1 just to make sure that we know what interval we are talking about elements of B 0 1 are called Borel measurable sets simply Borel sets. So, this C naught is the collection of all open intervals in 0 1 that is my collection that is like my collection C here. And obviously, there exists a smallest sigma algebra that contains all my open intervals by the theorem right it is uniquely defined that sigma algebra is called Borel sigma algebra on 0 1 it is a well defined sigma algebra according to this theorem it is uniquely defined sigma algebra correct. So, this Borel sigma algebra will contain a number of subsets of 0 1 it will definitely contain all the open intervals and it will contain compliments and countable unions of these open intervals right. And the sets in B 0 1 the elements of this sigma algebra are known as Borel measurable sets just like elements of any sigma algebra f are called f measurable sets. Now, this f is the Borel sigma algebra. So, we call it Borel measurable sets or simply Borel sets. So, Borel set is what it is an element of the Borel sigma algebra Borel set is the element of Borel sigma algebra. And the Borel sigma algebra is the smallest sigma algebra that contains all open intervals on 0 1. So, I think it is time for me to stop in a couple of minutes. So, now we have to wonder what kind of sets that this Borel sigma algebra contain right. So, it may be that. So, you do not know that this Borel sigma algebra is whether it is smaller than 2 power omega for example, right. I am this is I know that sigma c naught or B 0 1 or whatever is well defined as a sigma algebra, but it could be 2 power 0 1 itself right. In which case I have ended up with again the power set, but it. So, turns out this Borel sigma algebra is actually a much simpler sigma algebra than the set of all possible subsets of 0 1. In fact, this Borel sigma algebra has only as many sets as there are real numbers, but you know that 2 power 0 1 has the powers it has cardinality much higher right. So, cardinality of 2 power r right. So, this is actually a much smaller sigma algebra than the power set that can be shown. It is a non trivial result, but you can take it from me from for now that the Borel sigma algebra is a much smaller sigma algebra than 2 power omega, but the nice thing is it contains most sets of practical interest. In fact, it contains many sets which are quite bizarre Borel sets can be quite bizarre even something like the canter set right, which is this very peculiar set you know that is in fact, it can show as a Borel set. So, there are many subsets of 0 1 which are not Borel sets, but thinking of thinking of non Borel sets is very difficult it is there all very bizarre. So, many sets exist, but it is very difficult for you to think of a set which is not a Borel set. So, just mull over what kind of things this thing contains we will get back to discussing Borel sets in next class. No the C naught is set of all open sub intervals of omega all open sub intervals of omega. So, you say all open sub intervals all sets of the form open interval a b, where a and b are in 0 1 any other questions all right we will stop here.