1 00:00:19,369 --> 00:00:26,250 now let me define a another extension of the definition of ring and we call this structure 2 00:00:26,250 --> 00:00:39,390 as a sigma ring we can also write it in this fashion sigma ring so this is an extension 3 00:00:39,390 --> 00:00:44,320 of the definition of ring in the sense that a ring was closed under the operation of taking 4 00:00:44,320 --> 00:00:51,559 ah finite unions and differences if we change the finite unions to countably when unions 5 00:00:51,559 --> 00:01:02,050 then it becomes the definition of a sigma ring so a formal definition is a non empty 6 00:01:02,050 --> 00:01:13,990 class of subsets of omega so let me denote this class by say is script s is said to be 7 00:01:13,990 --> 00:01:24,560 a sigma ring if it satisfies the following two properties 8 00:01:24,560 --> 00:01:40,170 that given any two sets they are difference must be in the class and secondly if i consider 9 00:01:40,170 --> 00:01:53,090 any sequence then its union must belong to 10 00:01:53,090 --> 00:01:57,709 so usually you can see that the second one is a generalization from the definition of 11 00:01:57,709 --> 00:02:08,519 a ring because there we assumed only finite union so naturally the consequences that every 12 00:02:08,519 --> 00:02:28,170 sigma ring is a ring secondly if i have a ring and it is closed under 13 00:02:28,170 --> 00:02:43,609 closed under the formation of countable unions then it is a ring then it is a sigma ring 14 00:02:43,609 --> 00:02:56,379 ah as your seen in that in the definition of ah ring ah we assumed only closeness under 15 00:02:56,379 --> 00:03:01,709 the formation of unions and the differences however we could show that it is closed under 16 00:03:01,709 --> 00:03:07,969 the formation of symmetric differences the formation of intersections ah etcetera 17 00:03:07,969 --> 00:03:12,409 in a similar way if i am considering a sigma ring then it will be also closed under the 18 00:03:12,409 --> 00:03:20,670 formation of ah intersections so let us write down that a statement if i have a sequence 19 00:03:20,670 --> 00:03:38,889 a ah e n in s when we can write intersection e n n is equal to one to infinity as e minus 20 00:03:38,889 --> 00:03:52,629 union e minus e n n is equal to one to infinity where e denotes the set union n is equal to 21 00:03:52,629 --> 00:03:59,810 one to infinity now if is script s is a ah s is a sigma ring and if i am considering 22 00:03:59,810 --> 00:04:08,999 a sequence e n their then naturally the countable union of the sets belongs to the class s now 23 00:04:08,999 --> 00:04:17,269 e minus e n also belongs because this is the differences of that sets and therefore if 24 00:04:17,269 --> 00:04:22,030 i again take the complim ah the countable union that is again belonging to the class 25 00:04:22,030 --> 00:04:30,720 s and therefore e minus this is again in this so this belongs to s the now since this is 26 00:04:30,720 --> 00:04:36,710 n ah e this is a alternative representation of the intersection e n this means that every 27 00:04:36,710 --> 00:05:04,030 sigma ring is closed under is closed under the formation of countable unions ah ah the 28 00:05:04,030 --> 00:05:09,370 previous class we introduced the concept of the limit of a sequence of sets ah it was 29 00:05:09,370 --> 00:05:16,199 actually define ah as a limit superior of a sequence of sets limit inferior of a sequence 30 00:05:16,199 --> 00:05:23,330 of sets and if the two are equal then the limit exists now ah we had alternative representation 31 00:05:23,330 --> 00:05:29,050 of the limit superior and limit inferior in the form of countable unions of countable 32 00:05:29,050 --> 00:05:36,460 intersections are countable intersections of countable unions so now if by initially 33 00:05:36,460 --> 00:05:41,699 structure is a sigma ring and it is closed under the formation of countable unions and 34 00:05:41,699 --> 00:05:45,930 countable intersections therefore it is also closed under the operations 35 00:05:45,930 --> 00:05:51,241 of limit superior and limit inferior and therefore if the limit exist then under the operation 36 00:05:51,241 --> 00:06:11,379 of taking limits so you can mention that a sigma ring is closed under 37 00:06:11,379 --> 00:06:26,629 the formation of limit operations on the sequence of sets one more extension of the definition 38 00:06:26,629 --> 00:06:35,689 of ah ring are algebra r sigma ring is the so called structure called sigma algebra are 39 00:06:35,689 --> 00:06:51,129 a sigma field so let us define it in the following fashion sigma algebra are a sigma field so 40 00:06:51,129 --> 00:07:06,319 a non empty class of subsets of omega so let me define this class as a script f is said 41 00:07:06,319 --> 00:07:20,680 to be a sigma field if it satisfies that for a any given set its complementation will also 42 00:07:20,680 --> 00:07:33,009 be in the given class secondly for any sequence of sets the countable union must also be in 43 00:07:33,009 --> 00:07:38,650 the class so you can see that it is a generalization 44 00:07:38,650 --> 00:07:45,750 of the definition of a field as well as its generalization of the the definition of sigma 45 00:07:45,750 --> 00:07:51,949 ring its a generalization of the definition of field in the sense that in a field we are 46 00:07:51,949 --> 00:07:58,610 closed under the operation of taking complementations and unions but unions where taking to be finite 47 00:07:58,610 --> 00:08:04,150 here we have taken countable unions its an extension from the definition of a sigma ring 48 00:08:04,150 --> 00:08:10,189 in the sense that countable unions are there and the differences have been replaced by 49 00:08:10,189 --> 00:08:18,560 complementations so in some sense its one of the largest structures among the four structures 50 00:08:18,560 --> 00:08:35,640 that we have define so ah we can write the comments that every every sigma field is a 51 00:08:35,640 --> 00:08:52,959 sigma ring every sigma field is a field and therefore every sigma field is also a ring 52 00:08:52,959 --> 00:09:22,889 we can also say that a sigma ring closed under the formation of compliments is a sigma field 53 00:09:22,889 --> 00:09:48,980 a sigma ring containing omega is a sigma field a field closed under countable unions is a 54 00:09:48,980 --> 00:10:02,470 sigma field so among the four structures this sigma field are sigma ring is the most generally 55 00:10:02,470 --> 00:10:14,240 structure ah a related structure is that of a monotone class 56 00:10:14,240 --> 00:10:35,399 a non empty class of subsets of omega is called a monotone class 57 00:10:35,399 --> 00:10:59,260 if it is if for every monotone sequence e n in this class 58 00:10:59,260 --> 00:11:14,829 unit of e n belongs ah we prove that ah a monotone ring is a sigma ring and a sigma 59 00:11:14,829 --> 00:11:22,440 ring is a monotone class ah see we already said that in a sigma ring the limit operations 60 00:11:22,440 --> 00:11:28,060 are valid and therefore in the sigma field also the limit operations are valid 61 00:11:28,060 --> 00:11:35,399 so monotone class is a particularly structure ah which is something like in between a ring 62 00:11:35,399 --> 00:11:40,000 and the sigma ring are between a field and a sigma field however it is useful in the 63 00:11:40,000 --> 00:11:46,329 sense that give you in a class if i just look at whether the limits are their then monotone 64 00:11:46,329 --> 00:11:56,589 class is confirmed and in many operations that is what we finally need so let me ah 65 00:11:56,589 --> 00:12:13,440 prove the following theorem a sigma ring is a monotone class 66 00:12:13,440 --> 00:12:30,370 a monotone ring is a sigma ring to prove that a sigma ring is a monotone class we notice 67 00:12:30,370 --> 00:12:42,170 that a sigma ring was closed under the operations of taking infinite unions and infinite intersections 68 00:12:42,170 --> 00:12:47,140 and therefore the limit operations were valid therefore a sigma ring is naturally a monotone 69 00:12:47,140 --> 00:12:54,510 class because if a monotone sequences taken its union are intersection is the limit depending 70 00:12:54,510 --> 00:12:59,949 upon whether you have a ah monotonically increasing sequence or a monotonically decreasing sequence 71 00:12:59,949 --> 00:13:25,829 so the first statement is valid as a sigma ring is closed under countable unions and 72 00:13:25,829 --> 00:13:43,860 intersections ah to prove the second statement let us see in the following fashion 73 00:13:43,860 --> 00:13:59,649 let r b a ring and also a monotone class so since its already a ring we have to only 74 00:13:59,649 --> 00:14:05,029 show that countable unions will belong to the given class to prove that it is a sigma 75 00:14:05,029 --> 00:14:18,259 ring so let us consider a sequence a n in r now if i take say b n is equal to union 76 00:14:18,259 --> 00:14:26,089 of a i i is equal to one to n then it is a finite union of the sets in r and therefore 77 00:14:26,089 --> 00:14:35,149 it will belong to r for every n now the nature of this set b n is that if i take b n plus 78 00:14:35,149 --> 00:14:40,000 one then one more set will be coming so b n is naturally a monotonically increasing 79 00:14:40,000 --> 00:14:48,889 sequence of sets and limit of b n will become equal to union a i i is equal to one to infinity 80 00:14:48,889 --> 00:15:04,980 since r is a monotone class limit of b n will belong to r as r is a monotone class so r 81 00:15:04,980 --> 00:15:14,699 is a sigma ring ah this result is useful in the sense that if we want to create a sigma 82 00:15:14,699 --> 00:15:23,740 ring form a given class of sets then ah taking all the unions ah etcetera may be quite complicated 83 00:15:23,740 --> 00:15:29,899 where as if it is already a ring if we insure that the ah limits of the sequence of the 84 00:15:29,899 --> 00:15:36,310 sets is ah present in the given class then it will become a sigma ring so it is something 85 00:15:36,310 --> 00:15:43,550 like a ah you can say a method of obtaining generated sigma rings from a given class which 86 00:15:43,550 --> 00:16:03,139 is already a ring ah a somewhat simpler structure ah in this direction is a semi ring 87 00:16:03,139 --> 00:16:23,370 a non empty class say p of subsets of omega is said to be a sigma ring a said to be a 88 00:16:23,370 --> 00:16:40,310 semi ring if it satisfies the following properties that is if a set is there 89 00:16:40,310 --> 00:16:48,560 then e intersection f belongs to p that means it is closed under the formation 90 00:16:48,560 --> 00:16:55,079 of the taking intersections ah however a another property which looks somewhat different and 91 00:16:55,079 --> 00:17:04,670 the once which we have until now is that is e and f are any to subsets of omega n t and 92 00:17:04,670 --> 00:17:20,650 say one of them is a subset of the other one then there is a finite class c naught c one 93 00:17:20,650 --> 00:17:38,510 up to c n of sets in p such that e is equal to c naught subset of c one subset of c n 94 00:17:38,510 --> 00:17:52,880 which is equal to f and the success see would difference is of c i is must be in p i mention 95 00:17:52,880 --> 00:17:59,210 that it is a structure which is ah simpler compare to the structures that we define just 96 00:17:59,210 --> 00:18:06,400 now in the sense that i am not assumed the ah that union will be there our complementations 97 00:18:06,400 --> 00:18:20,120 will be there ah you can see here ah through an example if i consider say omega to be any 98 00:18:20,120 --> 00:18:36,900 set and p is say the class consisting of the phi set and the set of consisting of ah single 99 00:18:36,900 --> 00:18:49,409 ten sets for all x belonging to omega then this is a semi ring 100 00:18:49,409 --> 00:18:54,970 let see how if i take intersection of any two sets then that will be empty set if i 101 00:18:54,970 --> 00:18:59,340 take any two sets is most that is the subset of another one the second property is trivially 102 00:18:59,340 --> 00:19:06,200 satisfy so this is a semi ring now why i said that it is a structure which is ah more simpler 103 00:19:06,200 --> 00:19:11,860 in nature compare to a ring sigma ring field or sigma field because now you see this set 104 00:19:11,860 --> 00:19:18,659 p it is not satisfying properties of none of the previous structures for example unions 105 00:19:18,659 --> 00:19:25,250 of two sets are not there so it cannot be a ring sigma ring field are sigma field whereas 106 00:19:25,250 --> 00:19:32,400 it is a semi ring let me take another example suppose i consider omega to be the set of 107 00:19:32,400 --> 00:19:41,500 yield numbers and i consider p to be the class of intervals of the form a to be that means 108 00:19:41,500 --> 00:19:53,920 semi ah closed intervals i am taking the left closed and right open then this is a semi 109 00:19:53,920 --> 00:20:03,380 ring to see the structure of this suppose i am considering any two intervals of the 110 00:20:03,380 --> 00:20:19,120 form a to b now if i take say a one b one and say a two 111 00:20:19,120 --> 00:20:25,029 b two then if i take the intersection of this then if there of this form then the intersection 112 00:20:25,029 --> 00:20:34,289 of a one b one with a two b two is of the form a two b one which is again and interval 113 00:20:34,289 --> 00:20:40,529 of the same form suppose i take these to be disjoint then the intersection will be phi 114 00:20:40,529 --> 00:20:52,260 which is correspondent to the twice a is equal to b here similarly i may take this on this 115 00:20:52,260 --> 00:21:02,370 side say a two b two in that case the intersection will be a one b two which is again and interval 116 00:21:02,370 --> 00:21:12,300 of the same form so in all the cases the intersections will be existing in the class b now suppose 117 00:21:12,300 --> 00:21:20,180 i consider two intervals such that one of them is contained into another one so let 118 00:21:20,180 --> 00:21:26,220 me take a one b one contained in the interval a two to b two 119 00:21:26,220 --> 00:21:39,520 now if i look at this then i can consider n classes which are n sets ah here if you 120 00:21:39,520 --> 00:21:44,950 look at the second property of the definition of the semi ring then there is a finite class 121 00:21:44,950 --> 00:21:50,750 of sets in p c naught c one c n such that there is smallest of this is same as e and 122 00:21:50,750 --> 00:21:56,910 the largest of this is same as f such that they are difference is belong to p so if you 123 00:21:56,910 --> 00:22:04,900 look at this one i can always construct sets like this so this set itself can be i c naught 124 00:22:04,900 --> 00:22:15,490 this can be some set let me call its a ah c one to d one say c two to d two c three 125 00:22:15,490 --> 00:22:24,980 to d three and this is say c n to d n then each of them is a super set of the previous 126 00:22:24,980 --> 00:22:30,250 one the largest is equal to the bigger interval a two to b two the smallest is equal to the 127 00:22:30,250 --> 00:22:37,169 interval a one b one all of the sets are of the same form and therefore ah they are satisfying 128 00:22:37,169 --> 00:22:44,140 the property that they are in p if i consider the difference of any two ah 129 00:22:44,140 --> 00:22:53,510 successive sets so for example if i take c two a two ah d two minus c one d one then 130 00:22:53,510 --> 00:23:03,169 it is equal to now in this particular structure the difference is equal to ah c two d two 131 00:23:03,169 --> 00:23:14,250 so i am removing c one d one from here then it is becoming equal to ah c two 132 00:23:14,250 --> 00:23:24,350 c two c one ah so if i look at the ah definition of the semi ring then it is belonging to the 133 00:23:24,350 --> 00:23:51,030 class p so ah we are the following remarks phi belongs to p for every semi ring because 134 00:23:51,030 --> 00:24:00,110 i can take two sets we equal and if i take the differences then it will be there an equivalent 135 00:24:00,110 --> 00:24:28,550 definition of semi ring would be a non empty class p of subsets is a semi ring 136 00:24:28,550 --> 00:24:49,600 if i am t set is there for any two sets intersection must be there and thirdly if a and b belong 137 00:24:49,600 --> 00:24:58,159 to p then i should be able to represent a minus b as the union of a finite unions of 138 00:24:58,159 --> 00:25:06,620 sets in p so this is an alternative ah representation 139 00:25:06,620 --> 00:25:17,830 of the definition of semi ring ah a useful concept in this context is that of a generated 140 00:25:17,830 --> 00:25:25,260 class what is the generated class so given a class of sets if i consider the smallest 141 00:25:25,260 --> 00:25:33,270 ring containing c is smallest sigma ring containing c the smallest algebra containing c are the 142 00:25:33,270 --> 00:25:38,240 smallest sigma algebra containing c are the smallest monotone class containing c then 143 00:25:38,240 --> 00:25:45,720 it is called a generated ring it generated sigma ring a generated algebra are a generated 144 00:25:45,720 --> 00:25:54,190 sigma algebra ah now to see this suppose i look at say a set consisting of single time 145 00:25:54,190 --> 00:26:05,049 set a then phi a suppose i will say this is my c then this is the generated ring of c 146 00:26:05,049 --> 00:26:13,090 suppose i take this same thing and i take phi a a compliment and omega then this is 147 00:26:13,090 --> 00:26:24,260 a generated algebra from c ah since this is a finite class therefore 148 00:26:24,260 --> 00:26:29,990 this is also generated sigma ring are a generated sigma algebra to see that ah how this process 149 00:26:29,990 --> 00:26:38,279 becomes more complicated if i considered larger classes is that suppose i consider say class 150 00:26:38,279 --> 00:26:52,320 say a b now you can see in order to generate a ring out of this i have to consider the 151 00:26:52,320 --> 00:27:03,660 empty set the two sets they are union they are difference is as we have said its intersection 152 00:27:03,660 --> 00:27:10,510 will also be there its symmetric difference will also be there so now you can see that 153 00:27:10,510 --> 00:27:16,210 variety of sets b minus a must also be there because we can take the difference of these 154 00:27:16,210 --> 00:27:22,720 two so a generated ring is of this nature which 155 00:27:22,720 --> 00:27:27,270 is consisting of many sets now you see how the process will become even more complicated 156 00:27:27,270 --> 00:27:34,899 if i try to generate a ah field out of this if i consider a field generated form d then 157 00:27:34,899 --> 00:27:41,730 the number of sets is much more i will have to take phi a b then i have to take a compliment 158 00:27:41,730 --> 00:27:48,200 b compliment a union b then a compliment union b compliment can i have take a compliment 159 00:27:48,200 --> 00:27:55,179 intersection b compliment and so on so forth so the number of sets will be much more and 160 00:27:55,179 --> 00:28:00,240 this shows that this kind of a structures will be useful in the definition of probability 161 00:28:00,240 --> 00:28:06,960 because when we want to introduce the probability of different events when ah their simultaneous 162 00:28:06,960 --> 00:28:12,769 occurrence their differences their unions etcetera also will be needed in defining the 163 00:28:12,769 --> 00:28:21,769 probabilities so this type of definition will be extremely useful for ah ah for our 164 00:28:21,769 --> 00:28:22,219 thank you