Lecture 03: Fields and Generating sets for Sigma-fields

Welcome to this lecture. In the previous lecture, we had discussed about random experiments and the events that occur in a random experiment. More generally, we looked at the structure of these events that appear and we formulated these collections of events in terms of sigma fields. We had seen many examples of sigma fields and we had also used this bottom-up approach to construct examples. Now, in this lecture, we are going to continue the discussions on the structures on top of the collections of events. So more generally, as we have already talked about, we will take a non-empty set omega. For a random experiment, that will be our sample space, which is the collection of all outcomes. But in general, we are going to talk about a non-empty set and some collections of subsets of it. We are now going to look at other structures that appear on top of these special collections of subsets. So, as before, I will move on to the slides that are already been prepared. So, I start on the discussions on the slides. So, this is the definition. So, a collection script F of subsets of a non-empty set omega is called a field, if you have the empty set belonging to the field and F is closed under complementation as well as finite union. Just to compare with the structure of sigma fields that we have already seen, we already had empty set in the collection, but we had closure under complementation as well as countable unions. But now, we are talking about closure under complementation, but we are dropping the countable operations on union, we are only taking finite unions. So, as before for sigma fields, we are now going to look at structures on top of or properties on top of fields. So, what are these structures or what are these properties? So, if you take a field of subsets of omega, then the following statements hold. Again, these arguments are similar to whatever you have seen for sigma fields. So, the first property as listed here is that the whole set must be in the list must be in the field. So, again the idea is that you use the fact that the field is a closed under complementation and the whole set omega is nothing but the complement of the null set. Null set is already there. So, therefore, its complement must be there. The second property that we have seen similar to sigma fields once more is that F is closed under finite intersections. So, again as before you have to make use of the operations of complementation and finite unions to prove this property and the proof is extremely simple once more and it is left as exercise. Again, it is similar to whatever you have done for sigma fields. An important note, some authors prefer to use the term algebra and that is in our notion algebra is same as a field. So, the main idea is that for sigma fields, we had the similar term sigma algebra. Now, for fields we have using the similar term algebra. So, you would now ask what are now standard examples. The standard example first of all turns out to be sigma fields. So, remember all sigma fields that you get are closed under complementation as well as countable unions. But then you would like to verify whether this is closed under finite union also and that is easy to check because you can still take finitely many sets from the collection. Let us say a1, a2, an. You want to show that the finite union is there, but you can take this finite union as the union of a1, a2, an union with empty sets. Put empty sets and treat it as a sequence empty sets throughout after the n plus 1th step and then what you will end up having is that the union must be there in the sigma field which simply boils down to the finite union. This property we had discussed as a part of properties of our sigma fields. So, therefore, sigma fields are closed under finite unions and hence you would satisfy the requirements of the fields. And there are of course, closure under complementation is already there. Empty set belonging to the sigma field is already there as part of definition of a sigma field. So therefore, sigma fields are special structures which are automatically becoming a field. Now important example that not all fields are sigma fields. So, fields are somewhat less restrictive. So, you would expect to have more examples. So, here is the first example of a field which is not a sigma field. So, what we do? We first look at some specific collection of subsets of R. So, we denote it by script C. So, again script C is not any standard notation just for the example here we are discussing this. So, let us look at intervals of this type open A to closed B. So, on the left side it is open, on the right hand side it is closed. What are A and B? A and B run between minus infinity to plus infinity but A should be less than B. So, we are looking at such intervals which we call as left open right closed intervals. We would clarify what we mean when we put A equals to minus infinity or B equals to infinity. We would clarify that but let us say you understand what are these left open right closed intervals are then what we consider are finite disjoint unions of such things. Meaning if you have A1 B1 A2 B2 AN BN N many such things let us say then if they are pair wise disjoint take their union. So, that would be a typical element in this collection script C. So, let us say you have three such sets A1 B1 A2 B2 AN BN left open right closed if they are pair wise disjoint then take their union whatever the set you get that will be a set in our collection. Similarly you can extend it to any and many sets finitely many. So, let us now clarify what do we mean by left open right closed and what do we mean when A is minus infinity as B is infinity in this setup. So, a few clarifications about this definition we are now going to discuss. So, the clarification is like this. So, first of all the empty set is treated as a empty or a vacuous union of intervals. So, the intervals that we are looking at are left open right closed. We are saying that we are only going to consider finitely many unions of such sets but for a finite union we can also take a empty or a vacuous union. You don't take any sets then automatically as part of this description empty set belongs in this collection script C. So, whatever collections we are going to consider we are going to allow empty set billing there. So, that is the first thing. Next is the clarification about what happens for this left open right closed sets when A equals minus infinity or B equals plus infinity. So, the idea is simple. Suppose you are taking A equals to minus infinity. Now observe that all these sets that we are going to consider open A to close B must be subsets of the real line and the real line does not have the points minus infinity or the plus infinity. So, therefore whatever sets you are going to mean by minus infinity comma B open on minus infinity side or whatever then you are going to mean that these are specific subsets. So, for this case when the left boundary side the A value is minus infinity we therefore will take the set minus infinity open to closed B. So, that is easy to understand. So, for open A to closed B if A is minus infinity we simply look at the set minus infinity to closed B. But then similar statements would apply for the case B equals to infinity. So, again the idea will be that you are only going to consider A to infinity, infinity on the open side. So, we are not going to consider the point plus infinity as part of these sets because these sets are supposed to be within the real line and the real line does not contain the points plus or minus infinity. So, therefore for left open right closed sets we are going to consider such things. So, few more clarifications why we want to do it this way is this because we want to make this script C to be a field and for this we want the closure under complementation. So therefore, since we are aware that minus infinity open to closed B that set is in the collection script C its complement must also be there. So, therefore what we consider is that its complement if it is in there therefore we are asking that open B to open infinity is there. So, this complementation is being taken within the real line. So, as such we want these sets open B to open infinity to be inside the collection script C and for that to happen we want these sets to be treated as left open and right closed intervals. So, this is the clarification. So, if your right boundary point is plus infinity in that case what we incorporate is that the understanding is that we are going to write that specific interval with infinity open, but that is still left open and right closed. So, all of this is happening within the real line. So, we are taking complementations within the real line. Now what you can now try to verify is that with these structures and these notations at hand this collection script C above that we have described is a field. So, we are going to consider this thing. So, now this is a easy verification. So, setup for the complementation is already done, but you have to do much a bit more verification because the collection script C if we go back to the definition once more. So, definition says you have to take finite disjoint unions of left open right closed intervals like this. So, then you will take finite many such left open right closed intervals you would look at their complementation. So, how does it look like try to write it down you will immediately get the answer. Similarly, we have already described the empty set being in the collection script C or the only thing left to verify is the finite union possibilities and then you will also immediately get it because unions of left open right closed intervals if you take unions of two such things then either you get finite disjoint union or you simply get a left open right closed interval just try to verify this. So, if you have open A1 to closed B1 open A2 to closed B2 then their union either is a disjoint union of these two intervals otherwise if they intersect the union will simply become one big left open right closed interval try to check the calculation. So, then that is left as exercise for you. Now, what we started this discussion about is that we want to claim that this collection script C is a field, but not a sigma field. So, therefore, this should be an example saying that fields are much more general collections than sigma fields. So, to see that script C is not a sigma field what you observe that if you look at this open interval. So, now, we are not considering left open right closed, but we are considering open intervals. So, what are this open interval I am going to look at it is this open 0 to open 1. So, that interval I am going to look at. So, suppose I look at that interval now what is happening is that the since this is not right closed this open interval is not in the collection script C alright. So, that is as per the description. However, what you can do you can still write open 0 to open 1 that open interval as a countable union of sets coming from script C how. So, you look at such intervals on the right hand side written on the right hand side. So, open 0 to closed side, but on the boundary closed boundary right hand side will be 1 minus 1 by n plus 1 where n varies from 1 to infinity. If you look at such left open right closed intervals their countable union exactly turns out to be the open interval 0 1 try to check this it is very easy to verify alright, but then each of these sets open 0 to 1 minus 1 by n plus 1 closed. So, left open right closed this is in the collection script C, but their countable union open 0 to open 1 is not there in the script C alright. So, therefore, script C is not closed under countable unions. So, therefore, it cannot be a sigma field. So, this gives a explicit example of a collection which is a field, but not a sigma field. So, now there are several exercises which will help you understand these properties of a field much better. So, what are these exercises? So, please take your time working this out. So, suppose exercise 3 says if you take a countable infinite set for example, it could be the set of integers or set of natural numbers. So, for this such sets consider the collection script F of subsets we consider it this way. So, you look at subsets of the original set omega, but what are these properties of the subsets? So, either A is finite or its complement is finite. In either of these situations if A is finite or its complement is finite, I will put A in the list. I will put A in this collection script A. Now, if you take if this way you can now try to verify that it turns out to be a field. So, try to work this out. Now, here is some more structural properties involving fields. So, suppose you take a collection script D of some non-empty set omega of some subsets of non-empty sets omega such that it has this following properties that the empty set is in this collection script D. If A and B are two sets then their intersection is also in script D and the third property is interesting. It says that if you have a set in your collection then the complement can be written as a finite disjoint union of members of script D. So, if A is there you look at A complement. I am not claiming whether A complement is in the collection script D or not, but I would be able to write it as a finite disjoint union of members of D. So, there will be some sets let us say D1, D2, Dn whose disjoint union will turn out to be the A complement. So, for each A you can try to figure this out. So, if this you have these properties of script D then what you consider is a collection script F made out of properties of script D this way that you now consider this finite disjoint union of members of D. If you look at such sets then it so happens that script D turns out to be a field. So, this is a general property that you can use to construct fields, but for this you need a collection script D satisfying these three properties. This may look abstract to you, but here is a more concrete example for you. So, we are going to work out exercise 5. We are going to discuss this exercise 5 using exercise 4. What you can do? You can give alternative solution to exercise 2. What was exercise 2? Let us go back. So, in exercise 2 we had looked at this collection script D of left open right closed intervals and their finite disjoint unions. So, we said that that was a field on the real line. So, what we are going to say is this take omega the non-empty set to be the real line and script D as the collection of left open right closed intervals in R. What do you first try to do is that check that script D satisfies the properties as mentioned above meaning it should be closed under intersections and finite intersections and all these properties. Try to work this out. Once you have a script D for this specific case on the real line then check that this finite disjoint unions of members of script D exactly turns out to be the collection script C. And therefore, you apply exercise 4 to claim that since script E in this specific example of course, must be a field. So, therefore, script C which is equal to script E in this case must be a field. So, this is the alternative way of proving that script C is a field. So, when we try to prove script C is a field in exercise 2 we try to verify the actual properties of a field directly. Now, we are going via some actual construction behind in the background using some specific collection script D. So, that is what this exercise suggests. But then once you are looking at this collection script C we understood that we had to understand some properties involving this minus infinity and plus infinity. These points are not in the real line. So, we have to make sure that we understand this so that we can do algebra of this. So, this is our following note on the extended real number system. So, this will appear when you are working with limits or supremum or infimum of sets. So, if you are dealing with limits of sequences or supremum or infimum of sets all these operations sometimes will lead you to plus infinity or minus infinity. But the problem is this that on the real line you do not have these points plus infinity or minus infinity. So, what we do is that we use the extended real number system which we write as R bar which is nothing but the usual real line and you add these two extra points plus or minus infinity. So, now on this extended real number system R bar there is still some nice operations that you can still do standard algebraic operations. So, what are these? So, we are looking at the rules of arithmetic on the extended real line. So, on this in addition to the usual arithmetic that you already did on the real line which you already know you want to extend this to plus or minus infinity. So, what you do is that if you add or subtract infinity from a real number you get plus infinity or minus infinity. So, that is what this plus minus symbol represents. It means that if you add plus infinity you would get plus infinity if you subtract infinity you would get minus infinity. Then addition of two infinity is nothing but infinity and finally you can also subtract minus infinity from minus infinity itself. So, minus infinity and minus infinity that should represent minus infinity, but it is an important point that infinity minus infinity is left as undefined. So, this is the rules about addition and subtraction. Now this is the about multiplication that if you multiply a positive real number by plus infinity or minus infinity you get plus or minus infinity. If you multiply a negative real number by plus or minus infinity you end up with exchanging the signs. So, if you multiply a negative real number with plus infinity you get minus infinity and so on. And the important thing is that if the real number a is 0 then this is the important convention that infinity multiplied by 0 we will take it as 0. This is a very important convention. Moreover if you divide a real number by plus minus infinity you get 0 and then finally if you divide plus or minus infinity by a real number then you of course get the expected results this way. And importantly you do not divide infinity by infinity. So, these things are left undefined. So, again two things are left as undefined you do not subtract infinity from infinity and you do not divide infinity by infinity. So, as considered in the case of the real line you can still look at collection of subsets on the extended real line and you can still look at now left open right closed intervals and you do not have to worry about whether these points a and b are equal to infinity or not because here you are now considering them as subsets of the extended real line where plus infinity and minus infinity you are allowed to take. So, if you allow this then what happens is that you get this kind of sets where a and b could be directly put as minus infinity or plus infinity. So, if you look at such sets and look at finite disjoint union of such sets on the extended real line this will again make it a field. And then this is a clarification here now minus infinity closed to b closed that should be taken as a left open right closed interval. So, this set should also be included in the collection script C. So, for all this with all this setup you should try to check that script C must be a field, but it should not be a sigma field. So, this is again some repeat of arguments that we had already done for the fields case, but not a sigma fields case in the real line. Now, with this ideas at hand we are now going to consider going back to sigma fields and this is a very important construction of sigma fields a general construction. So, remember we had already discussed in the previous lecture that we saw this bottom up approach of constructing examples of sigma fields, but in this lecture what we are going to consider is a top down approach to construct more examples of sigma fields. So, what was this bottom up approach? We had the trivial sigma field the empty set and the whole set we added more sets and try to make it a sigma field. So, from the bottom which is the just a trivial sigma field we added more sets we go obtain bigger and bigger collection subsets which turned out to be sigma fields. So, from the bottom to up, but now we are going from top to down. So, what do I mean by this? So, on the other end of trivial sigma field as we had discussed earlier this is simply the power sigma power set sigma field. So, it consists of all possible subsets of the non-empty set omega. Now, if you look at the power set and now. So, that is on top on your hand what we do is that we try to remove sets from the power sets collection remove sets at judiciously. If you do it right then what will happen you will get smaller and smaller collections which will in principle could give you examples of sigma fields. And we will see now some exact operations that will allow you to choose this subsets in a significant way. So, that this once you choose them out in a certain way cross them off from the list from the power set list. If you cross them off then it will turn out that you can get this smaller collections which also turn out to be sigma field. So, this is a top to down approach. So, this idea is going through certain parametric collection of subsets of this non-empty set omega what do you need to do. So, take a collection script C of collections of subsets. So, this is just a collection of subsets we do not assume any properties or any structure on script C this is just a collection of subsets. Now, observe that the power set which we denoted by 2 power C omega this is a sigma field and this contains this smaller collection script C. So, all the sets in the script C are subsets of omega therefore, it is contained in the big list which is the power set. Now, consider all possible sigma fields script G this is a arbitrary sigma field on omega such that our collection script C is contained in script G. So, script G are bigger lists which contain the list script C. So, this contains the collection script C. Of course, we have already have such an example which is the power set sigma field, but we would like to look at all possible such sigma fields including the power set. Now, what we do we look at the collection script F which we denoted this way script F, but this should consist of all the common sets among all such G all such script G. So, what we do we first start with the collection script G which is containing our collection script C. Script G must be a sigma field this script G one example of this is the power set, but now what we look at we look at all the common sets among all such script G. So, we first list out all the sigma fields that contain our list script C look at all such G pick out the common sets in the lists. So, you might have 2 or 3 or more such lists of sigma fields. So, more such lists. So, look at all the common sets and create this collection script F. So, this in symbols we write this is the intersection of all this common sets. So, intersections among all this script G. So, this what are this script G these are must be sigma fields containing the script C. So, this is what sigma field this is what the script F looks like. By definition all sets in script C are common sets right because all sigma fields all such sigma field script G by definition contain elements or the members of script C. So, therefore, this script F also contains script C. So, therefore, all sets in the script C are contained there they are common sets. So, therefore, they are in the intersection. So, script C by definition is contained in script F all right, but then we now claim the script F that thus defined is a sigma field. How do you show this? So, we just verify the 3 properties of a sigma field for this collection script F. So, what you need to show first you need to show that empty set is in this collection for that you need to show that empty set is a common set among all the sigma fields, but since script G are sigma fields themselves they must contain empty set in their list right. So, therefore, what will happen is that empty set is in all of this script G which are sigma fields containing script C. So, therefore, empty set is a common set among all the sigma fields therefore, it is in script F that is by definition. More generally if you want to look at closure under complementation you take a set in script F, but then if A is a common set then A is in G for all such sigma field script G. So, therefore, it since each script G is a sigma field it is closed under complementation therefore, A complement is there in each of this script G. So, therefore, it is a common set alright. So, therefore, A complement is in script F. Similar argument will show you that the union of these things countable union of such sets common sets must also be a common set in the collections alright. So, therefore, countable union also must belong to script F. So, therefore, script F thus defined is giving an example of a sigma field. So, by construction again just to remind script C is contained in script F. So, all these sets that you already started with in script C they must be in script F. Now we make the definition we say script F is the sigma field generated by script C and refer to the sets in the collection script C as the generating sets of script F. So, you possibly get some extra sets then you started off with. So, you had a list of sets in script C, but you added more sets by a certain way you are looking at certain common sets among all such common script G alright. So, that will give you the necessary extra sets to make script F as the sigma field. So, this sigma field as said before we will call as a sigma field generated by script C alright and then elements or the members or the sets that are in the script C are said to be generating sets of script F and we will use this notation script F equals sigma within bracket script C to denote that script C generates script F alright, but then remember that script C is already contained in the sigma field script F and just rewriting this we just observe that C is script C is contained in the sigma field generated by script C great. Now sigma field generated by script C is a sigma field containing script C. So, therefore, what will happen is that the sigma field generated by script C by definition is equal to this intersection right. Therefore, sigma field generated by script C by definition is the minimal or the smallest sigma field containing script C. So, therefore, if you have any sigma field that contains script C this sigma field generated by script C these are the common elements and therefore, sigma field generated by script C is contained in any of this script G. So, therefore, what you are basically saying is that if script G is any sigma field such that script G contains script C. So, therefore, script G must contain sigma field generated by script C this is the minimal one containing script C. So, if you take any other sigma field then script G will contain sigma field generated by script C. Now we look at this properties of generating sets. So, if you have two such collections with one collection contained in the other then you can easily show that sigma field generated by script C 1 is contained in sigma field generated by script C 2. This is simply using the inclusions that we just discussed. This is just discussing the same properties that we have just mentioned. Take a minute to think about it you will immediately get it. But then I leave this question to you if script C is a sigma field on omega then can we conclude that script C is equal to the sigma field generated by script C itself. So, originally script C need not have any structure, but now we are saying what about taking script C and start off with that try to generate sigma fields out of it. So, if you can generate this that is using the general technique that we have just discussed what happens to this does it equal original script C itself. Now examples of such generated sigma fields is that if you take script C to be just the collection of one single subset capital A then you can show that sigma field generated by script C is nothing but this sigma field that we had already seen earlier. So, this is just the empty set then A, A complement and omega itself. So, you have this four sets collection which is a sigma field and that exactly is the sigma field generated by this subset capital A. More generally we are going to consider a very important collection of subsets which on the real line. So, if you take omega to be the real line consider the collection of all open sets. Now important point this collection of all open sets a priori does not satisfy the properties of a sigma field because you cannot have the properties of a sigma field here just try to check that, but then you can still generate a sigma field out of this open sets and what will happen is that you will end up with the Borel sigma field here and then we denote it by B subscript R. So, this is our notation that we are going to use. So, the generated sigma field out of all open sets in the real line. So, that is the collection script C you generate a sigma field whatever you get we refer to it as a Borel sigma field and for this special sigma field we introduce a notation which is script B underscore script R. So, this is called the Borel sigma field on R. Sets in the Borel sigma field that list that you end up having at the end these are called Borel sets or Borel subsets of the real line. So, you start off with the open sets generate a sigma field whatever extra sets or all the sets that you get including the open sets that you had started off with all the sets that you get are now called Borel subsets of the real line and most of our discussion in this course will use this Borel sigma field and one final property involving this generating collection subsets which we are going to see use of later on. So, if you have two such collections and with this property that script C1 is contained in sigma field generated by script C2, but sigma fields of generated by script C1 also contain script C2. So, this is the reverse inclusion options then you claim that the sigma field generated by script C1 and sigma field generated by script C2 these two must be the same that is the claim of this proposition. So, how do you go about proving this is using the properties that we have already discussed, but we shall see examples of such different collection script C which generate the same sigma field this we will see in the next lecture, but the verification shall be done through these applications of the proposition that we have just discussed. So, we are going to just check that script C1 is contained in the sigma field generated by script C2 and vice versa. If you verify that you will then claim that sigma field generated by C1 and C2 must agree. So, how do you prove this? So, you already know that sigma field generated by script C1 that is the minimal one containing script C1, but sigma field generated by script C2 is a general sigma field which contains script C1. Therefore, by note 15 sigma field generated by script C2 is a general sigma field and must contain the minimal sigma field generated by script C1. So, you get this first inclusion that you want, but you again use the reverse inclusion exchange the roles of C1 and C2. So, then again sigma field generated by script C2 is the smallest one containing script C2. Therefore, it must be contained in sigma field generated by script C1. So, therefore, you have both ways inclusion and you immediately claim the equality that you wanted to prove. This immediately follows from the relevant properties. So, now we have understood some structures of sigma fields and fields. We have used some examples and we have used some nice techniques bottom-up approach and a top-down approach in constructing many such examples. We are going to look at now very concrete properties of one of these sigma fields which we introduced a few minutes back which is the Bodele sigma field. So, we are going to discuss that in the next lecture.