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now let me define a another extension of the
definition of ring and we call this structure
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as a sigma ring we can also write it in this
fashion sigma ring so this is an extension
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of the definition of ring in the sense that
a ring was closed under the operation of taking
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00:00:44,320 --> 00:00:51,559
ah finite unions and differences if we change
the finite unions to countably when unions
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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
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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
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00:01:13,990 --> 00:01:24,560
a sigma ring
if it satisfies the following two properties
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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
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00:01:40,170 --> 00:01:53,090
any sequence
then its union must belong to
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00:01:53,090 --> 00:01:57,709
so usually you can see that the second one
is a generalization from the definition of
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00:01:57,709 --> 00:02:08,519
a ring because there we assumed only finite
union so naturally the consequences that every
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00:02:08,519 --> 00:02:28,170
sigma ring is a ring secondly if i have a
ring and it is closed under
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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
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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
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00:02:56,379 --> 00:03:01,709
the formation of unions and the differences
however we could show that it is closed under
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00:03:01,709 --> 00:03:07,969
the formation of symmetric differences the
formation of intersections ah etcetera
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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
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formation of ah intersections so let us write
down that a statement if i have a sequence
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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
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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
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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
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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
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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
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00:04:17,269 --> 00:04:22,030
i again take the complim ah the countable
union that is again belonging to the class
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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
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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
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00:04:36,710 --> 00:05:04,030
sigma ring is closed under is closed under
the formation of countable unions ah ah the
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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
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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
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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
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00:05:23,330 --> 00:05:29,050
of the limit superior and limit inferior in
the form of countable unions of countable
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00:05:29,050 --> 00:05:36,460
intersections are countable intersections
of countable unions so now if by initially
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00:05:36,460 --> 00:05:41,699
structure is a sigma ring and it is closed
under the formation of countable unions and
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00:05:41,699 --> 00:05:45,930
countable intersections
therefore it is also closed under the operations
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00:05:45,930 --> 00:05:51,241
of limit superior and limit inferior and therefore
if the limit exist then under the operation
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00:05:51,241 --> 00:06:11,379
of taking limits so you can mention that a
sigma ring is closed under
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00:06:11,379 --> 00:06:26,629
the formation of limit operations on the sequence
of sets one more extension of the definition
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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
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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
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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
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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
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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
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00:07:33,009 --> 00:07:38,650
the class
so you can see that it is a generalization
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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
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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
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00:07:51,949 --> 00:07:58,610
closed under the operation of taking complementations
and unions but unions where taking to be finite
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00:07:58,610 --> 00:08:04,150
here we have taken countable unions its an
extension from the definition of a sigma ring
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in the sense that countable unions are there
and the differences have been replaced by
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00:08:10,189 --> 00:08:18,560
complementations so in some sense its one
of the largest structures among the four structures
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that we have define so ah we can write the
comments that every every sigma field is a
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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
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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
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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
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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
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00:10:02,470 --> 00:10:14,240
structure ah a related structure is that of
a monotone class
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a non empty class
of subsets of omega is called a monotone class
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if it is if for every monotone sequence e
n in this class
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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
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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
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00:11:22,440 --> 00:11:28,060
are valid and therefore in the sigma field
also the limit operations are valid
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so monotone class is a particularly structure
ah which is something like in between a ring
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and the sigma ring are between a field and
a sigma field however it is useful in the
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sense that give you in a class if i just look
at whether the limits are their then monotone
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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
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00:11:56,589 --> 00:12:13,440
prove the following theorem
a sigma ring is a monotone class
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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
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that a sigma ring was closed under the operations
of taking infinite unions and infinite intersections
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00:12:42,170 --> 00:12:47,140
and therefore the limit operations were valid
therefore a sigma ring is naturally a monotone
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00:12:47,140 --> 00:12:54,510
class because if a monotone sequences taken
its union are intersection is the limit depending
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upon whether you have a ah monotonically increasing
sequence or a monotonically decreasing sequence
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00:12:59,949 --> 00:13:25,829
so the first statement is valid as a sigma
ring is closed under countable unions and
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00:13:25,829 --> 00:13:43,860
intersections ah to prove the second statement
let us see in the following fashion
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let r b a ring and also a monotone class
so since its already a ring we have to only
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show that countable unions will belong to
the given class to prove that it is a sigma
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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
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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
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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
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00:14:35,149 --> 00:14:40,000
one then one more set will be coming so b
n is naturally a monotonically increasing
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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
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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
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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
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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
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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
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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
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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
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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
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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
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00:16:23,370 --> 00:16:40,310
semi ring if it satisfies the following properties
that is if a set is there
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00:16:40,310 --> 00:16:48,560
then e intersection f belongs to p
that means it is closed under the formation
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00:16:48,560 --> 00:16:55,079
of the taking intersections ah however a another
property which looks somewhat different and
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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
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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
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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
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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
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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
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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
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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
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set and p is say the class consisting of the
phi set and the set of consisting of ah single
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00:18:36,900 --> 00:18:49,409
ten sets for all x belonging to omega then
this is a semi ring
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let see how if i take intersection of any
two sets then that will be empty set if i
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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
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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
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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
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p it is not satisfying properties of none
of the previous structures for example unions
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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
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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
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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
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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
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00:19:53,920 --> 00:20:03,380
ring to see the structure of this suppose
i am considering any two intervals of the
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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
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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
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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
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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
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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
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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
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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
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00:21:12,300 --> 00:21:20,180
i consider two intervals such that one of
them is contained into another one so let
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me take a one b one contained in the interval
a two to b two
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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
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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
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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
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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
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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
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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
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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
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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
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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
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00:22:37,169 --> 00:22:44,140
the property that they are in p
if i consider the difference of any two ah
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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00:25:25,260 --> 00:25:33,270
ring containing c is smallest sigma ring containing
c the smallest algebra containing c are the
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00:25:33,270 --> 00:25:38,240
smallest sigma algebra containing c are the
smallest monotone class containing c then
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00:25:38,240 --> 00:25:45,720
it is called a generated ring it generated
sigma ring a generated algebra are a generated
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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
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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
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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
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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
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00:26:29,990 --> 00:26:38,279
becomes more complicated if i considered larger
classes is that suppose i consider say class
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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
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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
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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
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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
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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
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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
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00:28:12,769 --> 00:28:21,769
probabilities so this type of definition will
be extremely useful for ah ah for our
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00:28:21,769 --> 00:28:22,219
thank you