 So, for this long, long, long back people try to put all these probability theory in a very, very systematic manner, so that all of us are on the same page when we are talking about these term things and also they lead to kind of intuitive properties that we want them to satisfy. So, that is given by something called some one called like Kolmogorov you might have heard this name. So, he built this all very systematically by introducing this notion of what we call as probability space and this is, so to do this he introduce a triplet called denoted as omega script f and p. Omega is same as earlier, it is the event space, sorry sample space, so what is the difference between sample space, event space, did we say that the same right, like I mean this is a collection of, so okay, so that, so what is, so I just by mistakenly said sample space, sorry event space, but I have only introduced this as sample space, so this what does the sample space constraints, it contains only possible outcomes, event space could be collection of events and that is where this f come to picture, so now it is not necessary that every subset of a sample space is an event of interest to you. Okay, I mean for example, let us say two guys, they are making some decision, let us say just making s and no decision, one guy says s no or other guy also has to say s and no and these guys are just like dead opposite guys, I mean they will never say the same thing. Then you already know that s s and no, no option is out of the picture, right, you do not want to consider that event at all, so and in that your events of interest are only now restricted, it is not that all possible subsets of your sample space okay, so that is there, now this will define what is that possible set of events that you are interested in and now once you restrict yourself to possible set of events, now on that you want to assign the probability that that outcome is going to happen and that is what the third term is going to define, okay, particular event that is coming from this event space, I am going to make it more precise, so let now this is your sample space, so this always happens, right, like do you see why this should be done, like why this, I will not always consider all possible subsets of sample space as event, why I will be only taking only some of the events, so you can imagine, right, like when you are going to do an experiment based whatever the outcome and when you are going to do this experiment, let us say two or three of you are going to do this or whatever you are individually doing, you know that some outcomes are not possible, even though that outcome is there in the sample space but when you are looking at that particular event that may not happen, so can anybody think of a example that you have seen in your daily life where this can arise, I do not know, let us take one hypothetical example like let us say you are reaching airport and your friend reaching airport and let us say you are another close relative reaching airport, what are the possibilities all three of you reach, none of you reach, one of them reach, two of them reach, like that, right, there are so many possibilities but you know that one guy is already in the airport, right, so that guy is never going to be, so that guy is already in the airport, right, that possibility that that guy is not airport, all of this option is already eliminated. So, there are always such among all the possibilities such combinations can be eliminated, right, so that is why you want to restrict yourself to some events which are not which is not necessarily always a subset of your omega, yeah, maybe you can think of better real life example, it is not on top of my mind now, okay. So, fine and this one we are going to call it as event space, this is basically set of subsets of, we are going to call it as event space and this last term we are going to call it as probability function or probability measure, okay. Now, let us see, so the script of here is basically collection of events, right, this is an event space. Now, as I said it could contain either all set of subsets of, all subsets of your sample space or it could contain only some subsets of your sample space, okay, but how that should be like if I want to do a consistent modeling, how this f event space should be, so for that we are going to make some axioms that is going to say how this f is supposed to be, okay, so for that we are going to say something like, we are going to call event space, okay, event space. So, let me write the properties for this and let us then discuss whether they are natural or you expect this thing to be at least as a necessary to happen. So, we are going to say some event space f to be sigma algebra if it satisfies the following, first thing I want is this omega to be included in my event space that is entire collection of my outcomes should be included in f and then if some event if belongs to f, then a complement belongs to f, f is not, no, f is a collection of subsets of omega, okay, so this is the omega, that means f is a set of sets, you understand this, that is what I said, right, set of subsets of omega, yeah, you can think of power set that is the largest possible event space you can get, right. So, okay, before I write this maybe an example is good here, okay, let us take our trivial thing omega equals to h and t, what could be an event space, it could be simply h, that is it, only an element or f could be h and t itself like it has the both elements or maybe just has f equals to t or it could be null set, anything is possible, right, that is what I am saying, I am going to focus on certain sigma algebras, I want, now I am telling you what are the properties of that event space, okay, what is the properties, I mean when I call it a sigma algebra, so first thing I am saying omega, omega is itself a subset of itself, right, omega is a subset of omega, so it can be a possible candidate in f, so I am saying, okay, when I say sigma algebra that omega belongs to f, if any, if suppose let us say some subset of omega, let us call it a belongs to f, its complement also belongs to f, then, okay, so in this case it makes more sense to write them as, so okay, fine, so this should be like this is one, this is just one and maybe this is one or other possibility here just to make it more clear is, it could be like h and t and just h, all of you get this, so I mean, so earlier this is what now the distinction, when I say it in a flower bracket, this itself is a set and here this is a set, this is a set and now this is f is a collection of these two sets, so a is such a subset, right, because this is a one element in f, one element in f means this is one subset of omega, is that clear, yeah, set you call a b or b a, how does it matter, it is just a collection of sets, yeah, you just say 5 people, you do not say who is who, okay, that is all right, order, I mean if you want to say order, you need to define what is order, right, when you say order, okay, this guy is greater than this, that guy is greater than this, order, we are not getting into, we are just writing, this is one set, this is one set and the third point we want is, okay, so we want is if a and b belongs to this f, their union also belongs to f and as an extension to this, also if a1, a2, 0, sequence of then, so okay, so I am saying if I have a collection of subsets, which I am going to call event space and I am going to call that event space, particular event space to be sigma algebra, if the following holds, so first thing is omega belongs to f, so do you think this is natural, I should include omega to be in event space, so simple question is okay, let anything happen, that means enter omega, right, so let omega be in f, okay and now if you say some event happens, there should be some meaning to that event not happening also, right, so that is why if a belongs to a, a complement belongs to it, if you say a can happen, b can happen, then you should be able to answer whether both can happen and that is what a union, b that has to be there in scripted and also you like that instead of just 2 or like finite number of unions to be in f, you can ask, okay if I have a sequence of events from my script f, all this a1, a2, they belong to f, then I want their union to be also in script f, so this is all right, when I am conducting trying to model something, some random thing and if I want to define the set of outcomes, in the set of outcomes, I would like the outcome omega to be included, if a belongs its complement to be included and this unions should also be included, okay let us take again, let us go back to our H and T, right, these are just outcomes, I said these are all possible subsets, right, these are collection of subsets which I called them, what is the best possible, what is the largest subset of these sets, number of, what is the largest number of subsets from the sets I can get, 4, right, what are those, H, T, H, T and null set, that is like a power set, right, but I had here considered only some subsets here, that is what you want to know like that is, so okay fine, let me write it, if there is one possible is f could be H, T, H, T and null set, these are the only possible subset of omega, right, I cannot add anything more here, this is the largest, but why should I look for the largest, I could also look for the smaller than this, in this case this could be one possibility, this could be one possibility, this could be, but all of this need not be sigma algebra, okay, now let us say, if I am going to define f to be like this for this omega, is this the sigma algebra, right, because now just, best thing is just go and apply plainly these axioms, so this is why we are going to call them as axioms, but because these are fundamental things which you need to hold, and these are definitions, when we are going to talk something sigma algebra, the only thing that is common to both of us is these definitions, okay, now let us take another H and T, if this is sigma algebra, so right away the first condition is violated, right, yeah, meaning of sigma algebra is implicit definition, I am telling this is what then you are asking what is that, so okay, telling it in other ways, sigma algebra is some sensible set of events, I want such that they satisfy some property, okay, that means reasonable sense, that is what I said, right, it is something which should include all possible outcomes as one possibility, if something happens naturally the question you are going to ask is what if that event does not happen, so both happening and not happening should be there, and the second condition is what that, so I am going to say, okay, that happened, this happened, can both happen, this is what you guys are always asking, right, I mean, yeah, if I did that course, can I do this course, if I drop that course, can I do this, if I want to shift it, can this be possible, I mean all permutation combination can happen, but as we all most of the time reject that, right, they are only going to take some possibilities, and this is what those some possibilities we are going to define, okay, so this is not a sigma algebra because it does not satisfy the definition, suppose I take a simple thing in this, so what is the simple thing, this is, is this a sigma algebra, yes, quickly apply for all the three, all the three condition, fine, right, it satisfies, omega is anyway there, if a, suppose if I take this to be the a, a complement is phi that already belongs, if I take a to be phi, phi complement is omega, so we are going to talk complement with respect to the sample space, so complementation is happening with respect to the sample space, so when I say omega equals to 1, 2, 3, 4, 5, 6, and let us say a equals to 1, 2, what is a complement, 3, 4, 5, 6, suppose now I redefine my omega to be just 1, 2, 3, 4, 5, if I take a to be 1, 2, what is the complement, so just this guy, so always my complementation is with respect to what omega I am talking about, okay, if my omega changes, my complement can also possibly change, and yeah that is the exact thing, my complementation is with respect to my event space, okay, these are like, this is a very toy model, right, we are not getting that more leg space here to construct a nice event spaces which are not sigma algebra, let us take bit the dive problem, so in this let me construct a simple thing, so one possible I am going to take is, so first I will going to include omega that entire thing, I will take 1, 2, 5 like this, and let us say I will ask you, give me one element here that will make this f sigma algebra, we just want to take the complement of 1, 2 to make it any other elements required, suppose I have added one more thing, now tell me how many terms I should include in this, so that it becomes sigma algebra, what are those, so okay fine, so now I have these elements this, so now tell me what all the things I should include in this, complements of this, okay, 3, 4, 5, 6, tell me one by one, so first go systematically, right, first thing is their union should be there, so 1, 2, 3, and then it is complement, okay, so I have the complement of this, now what is this, I have to bring a complement for that, right, what is that value, enough or I need more, more what, omega, right, so the union of this should be there, right, it is already there, okay, we are done, done, so no more addition is required, right, so fine, so what we are doing is we are just sticking to the definition of this, so that we are, we call it as sigma algebra, okay fine, what we just said is, in this condition, in this axioms, basic axioms, you will see that these are the bare minimum things I need in my event space, so in this I did not say that if a and b belongs to script f, a intersection b belongs to script f, do I need to explicitly say in this or that is implied, so a, b belongs to script f, I know that their union belongs to script f, suppose f is a sigma algebra, does a intersection b belongs to script f, why, okay, tell me a union b belongs to f, this implies then b complement and belongs to f, so does this imply, does this imply a intersection b belongs to f, okay, start wherever you want, what do you want to start with, fine, then I want to reach here, you want to take the complement, okay, I mean union, then you take, so if this belongs, I know that this complement has to belong to this, then it is, then once I have this, I also know that a complement union b complement belongs to f and you can see that this is nothing but a intersection b and once you have this, similarly you can go and extend, if you have a sequence of elements in f such that their union belongs to f, you can show that their intersection do also belong to f, same set of arguments, okay, so make sure that you work out that, okay, okay, so last thing, so we are going to now revisit what I have already defined p function, now I am going to make, define this, so p of, now this p is defined on f, see the way we have defined this triplet, omega is a sample space, f comes from this sample space and now this p is defined on this event space, once you give me this event space, I do not care what is your sample space, to define my p, so we have already written this, let me rewrite p of a is equals to 0 for all, it equals to f, okay, let us see now what is the implication of? Normal probability and probability is measured on f, so there is not any difference, we are just making this the same whatever we have defined earlier, now we are trying to define it on my event space, right, earlier I have just defined it on any possible event, right, you just take an event, now I am saying that fine, I do not need to define on any possible event, I am going to take this bit more systematically for a given random experiment, I am going to represent it using these 3, omega, script f and p, omega is out, outcome is the sample space and script s is possibly the events of my interest and on that this probability is now defined. And now it is the same thing, earlier a was any event, now I am making it more precise to make this replacement clear that this probability is now defined on the elements of script f, okay. Now, from this, is it true if p is a probability function, we are not going to use this word measure, measure is for a bit more abstract, more general things, I mean in the space we are looking at let us simply keep calling it probability function or a probability function defined in this fashion, is it true that probability of a, is a probability function, it is defined, this probability function is defined on an f, right, f includes null set, so I should know what is the value this probability gives on the null set, what is this value is going to be, can you show this or if you have to show this what should be the argument, okay, let us take in this case I know a, let us take a to be omega itself, omega belongs to f, right and by this if omega belongs to script f, I know omega complement also belong to script f, so omega is equals to omega union, omega complement, right and in the space of omega what is omega complement, it is a null set by definition and this omega null set this joint, yes, right, like there is nothing about this, now what is this I can, now how can I conclude, why is that because of my third part, so this concludes that my p of e is equals to 0, so do you expect p of a plus p of any thing to be 1, so can we, is that, is this necessary that I have to include this in as a fourth condition or this is be implied from these three conditions, how, so what is omega, right and is a and a complement are mutually exclusive, so then we are done, right and we know this guy is 1 by definition and now suppose let us say a, b belong to script f and let us say a is a subset of b, so do you expect the probability p to give more value to this event or this event, more value, you expect it to give more value to b, right that is, so do you think I should include this in this condition, so this is natural, right if we want this, if one event has more elements, this probability of likelihood is more, right, should I include this here or, so this is a requirement, right, I like this to be there, like I want p to satisfy this, is this implied by this condition or I have to include this, why, how, construct, so what can one of you say what could be the arguments to show this? b minus a intersection b is another set, b minus a intersection b, b minus a intersection b, ok I will just write whatever you want to say, b minus a intersection b, what is this, let us to make our life simple, let us take this to be a set, b bigger one and let us take a to be something contained in it, yeah, so I mean there could be many possibilities, can you think come in, I want you to give the full thing, ok fine let us try to construct, so the thing we know when you want to apply this, I want to like somehow bring in mutually exclusive sets and complement, so let us see, let us see if I, I know that this set a and whatever that in this, that disjoint, right, how can I represent the guy in this b minus a, ok you tell, is this correct, what is this b intersection a complement will give me, the shaded area only, right. Now, how can I write probability of b, how can I write b, a union, right, now let us call this c, some set, right, does c belong to f, a y, does c belong to f, why? There are two things that belong to f. So, I assume a belongs to f, so is a complement belongs to f, I assume b belongs to f, so both of this belong to f, so this intersection belong to f, I said intersection of two element belong to f also belong to f, right, so this c belongs to f. So now, I have this, I am going to just write it, I have this b written as a union c, I have argued that both are disjoint and both belong to f and on both of this I know p is defined. Now, what is this, I know p of b is equals to p of a plus p of c, why this? My third property, now with this I am done, now I go back to this and I know that p of c has to be non-negative, right, like it has to be greater than or 0, then it must be the case that pb is greater than or equals to p, so like this you can, these are like, that is why like these are called axioms, these are very, very elementary conditions I required, they are going to imply lot of things. So, they imply many things which I am going to just write and I expect you to work them out and some of them will be also in your assignment, so these are three, maybe like fourth is, so earlier I said that I expect this property to hold intually, but you can show that if these three conditions holds, this is indeed true for any a, b belong to script f. And you can generalize this by taking, yeah, other things we have derived and you can derive many more properties which I will skip. So, some of these more properties that you can derive out of these axioms, you will see in the tutorial session, yeah, okay, the last one. You can also visualize all this through this Venn diagram, right, I mean you should, whenever you, it is possible just draw Venn diagram that will help you to understand all these relations instead of memorizing the Venn diagram should help. Okay, so let us stop here.