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Unformatted text preview: Chapter 2 A modicum of measure theory 2 February 2004: Modi f cation of Section 2.11. *1. Generating classes of functions Theorem < Dynkin.thm > is often used as the starting point for proving facts about measurable functions. One f rst invokes the Theorem to establish a property for sets in a sigma f eld, then one extends by taking limits of simple functions to M + and beyond, using Monotone Convergence and linearity arguments. Sometimes it is simpler to invoke an analog of the system property for classes of functions. < 1 > Definition. Let H be a set of bounded, realvalued functions on a set X . Call H a space if: (i) H is a vector space (i) each constant function belongs to H ; (ii) if { h n } is an increasing sequence of functions in H whose pointwise limit h is bounded then h H . The sigma f eld properties of spaces are slightly harder to establish than their system analogs, but the reward of more streamlined proofs will make the extra, onetime effort worthwhile. First we need an analog of the fact that a system that is stable under f nite intersections is also a sigma f eld. Remember that ( H ) is the smallest  f eld on X for which each h in H is ( H ) \ B ( R )measurable. It is the  f eld generated by the collection of sets { h B } with h H and B B ( R ) . It is also generated by E H : = {{ h < c } : h H , c R } ....
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This note was uploaded on 11/21/2009 for the course STAT 330 taught by Professor Davidpollard during the Spring '09 term at Yale.
 Spring '09
 DavidPollard
 Probability

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