lambda-space

# lambda-space - Chapter 2 A modicum of measure theory 2...

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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 &lt; Dynkin.thm &gt; 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. &lt; 1 &gt; Definition. Let H be a set of bounded, real-valued 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, one-time 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 &lt; 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.

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lambda-space - Chapter 2 A modicum of measure theory 2...

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