Sheet8 - Statistics 330/600 Due Thursday 24 March 2005...

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Unformatted text preview: Statistics 330/600 Due: Thursday 24 March 2005: Sheet 8 *(8.1) Let ν and µ be finite measures on a sigma-field of subsets of X , with ν ¿ µ . Let T be an A \ B-measurable map into a set T equipped with a sigma-field B . (i) Show that T ν ¿ T µ . Write g for the density d ( T ν)/ d ( T µ) . (ii) Let ν and µ denote the restrictions of the two measures to the sigma-field σ( T ) . Show that g ◦ T is a version of the density d ν / d µ . *(8.2) Suppose A is a sub-sigma-field of A . Let H : = L 2 ( X , A ,µ) and H : = { h ∈ H : h is A-measurable } . Let H denote the closure (for the L 2 (µ) distance) of H in H . (i) Show that for each h in H there exists an h in H for which µ { x : h ( x ) 6= h ( x ) } = 0. (ii) For each f in H show that there exists an f in H for which µ( f F ) = µ( f F ) for every F in A . (8.3) Let ( X , A ,λ) be a measure space with λ a sigma-finite measure. Let T be an A \ B ( R k )-measurable map from X into R k . The natural parameter space is defined as...
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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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