Sheet5 - ∈ E i for each x in X(ii Show that 1 c = x 1 x 2...

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Statistics 330/600 Due: Thursday 17 February 2005: Sheet 5 *(5.1) Suppose E i is a collection of subsets of X i with X i E i , for i = 1 , 2. Define E 1 × E 2 : = { E 1 × E 2 : E 1 E 1 , E 2 E 2 } Show that σ ( E 1 × E 2 ) = σ( E 1 ) σ( E 2 ) , as sigma-fields on X 1 × X 2 . Hint: Show that A 1 : = { A σ( E 1 ) : A × E 2 σ ( E 1 × E 2 ) for each E 2 E 2 } is a sigma-field. Then what? *(5.2) Let X be a set equipped with a countably generated sigma-field A . That is, A = σ( E ) where E = { E i : i N } . Suppose also that { x } ∈ A for each x in X . Without loss of generality, suppose E is stable under complements. (i) Show that { x } = ∩{
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Unformatted text preview: ∈ E i } for each x in X . (ii) Show that 1 c : = { ( x 1 , x 2 ) ∈ X 2 : x 1 6= x 2 } = ∪ i ∈ N E i × E c i ∈ A ⊗ A . (iii) Let (Ä, F , P ) be a probability space. Suppose X : Ä → X is an F \ A-measurable map with distribution P . Suppose X is independent of itself. Show that there exists some x ∈ X for which X = x almost surely [ P ]. Hint: What do you know about P ⊗ P 1 c ?...
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