slides_1_probability

# 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
( 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
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 -field is closed to find 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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