Is a sequence of sets with a j ℱ j 12 then j a j

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is a sequence of sets with A j , j 1,2,. .. then j A j is closed under countable unions .) Given the above axioms, we can derive several additional features of a -field: 22
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( 4) . This follows by ( 1) and ( 2) because c . ( 5) is closed under finite unions: A , B A B . Follows from ( 3) and ( 4) because A B A B ... .... 23
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( 6) is closed under finite and countable intersections. Follows from ( 2), ( 3), ( 5), and DeMorgan’s Laws. For example, j 1 A j j 1 A j c c , which shows that if each A j then so is their countable intersection. ( 7) If A , B then A B ( is closed under set differences.) This follows easily because A B A B c and if A , B then so is B c and therefore so is A B c . 24
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EXAMPLE :(i) , is the smallest -field containing . (ii) The set of all subsets of is (obviously) the largest -field containing . (iii) If A is any subset of , then , A , A c , is the smallest -field containing the set A . When is a nonempty set and is a -field on , we say that , is a measurable space . For our purposes, this means that we will be able to define a probabilities for events in that satisfy certain, intuitive properties. 25
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Let G be a set of subsets of .The -field generated by G , denoted G , is the smallest -field containg G . In principle, one can list all elements of G and then apply all of the operations under which a -fieldisclosedtofind G . In practice, this is very difficult when G is large. Typically, one describes the kind of sets included in G . Eventually, because we study data, we will be interested in outcomes that are numbers of groups of numbers. So we want probability measures defined on , the set of real numbers. 26
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This is a case where the set of all subsets of is “too large” to define a probability. We need a -field that contains all the events we care about but is not “too large.” The Borel - field fits the bill; it is often denoted B or B , where the latter is used when the underlying sample space needs to be emphasized. Several ways to characterize B . Usually it is defined to be the -field generated by open intervals a , b for a , b , a b . 27
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It follows from the definition of a -field that B includes the intervals − , b and a , for any real numbers a and b . Why? B includes n , b for all integers n 1,2,. .. andso, because a -field is closed under countable unions, n 1 n , b B But it is easy to show that n 1 n , b − , b .
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is a sequence of sets with A j ℱ j 12 then j A j ℱ ℱ...

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