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# h6 - Z that is as n → ∞ with n ∈ Nr Z 6b Do...

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Probability II MAP6473 4810 Homework-6 Prof. JLF King 4May2010 Reading. Please read the ergodic theory chap- ter of Billingsley. Notation in force. For sets E,A N , say that E eventually includes A if E A [ N .. ) for some suﬃciently large N . 6a: Suppose that A 1 ,A 2 ,... are zero-density subsets of N . Then there exists a zero-density set E which eventually-includes each A j . [ Hint: Think Cantor diag- onalization.] Consider a sequence ~ X of non-negative reals. For each ε > 0, let N ε := { n | x n > ε } . For each ε , suppose that each N ε is a zero-density set. Prove that there exists a zero-density set Z N so that x n 0 oﬀ of
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Unformatted text preview: Z ; that is, as n → ∞ with n ∈ Nr Z . 6b: Do Billingsley’s problem, 24 . 7 P. 326. Here ( T : X, X , P ) is a mixing transformation. A fnc δ : X → [ , ∞ ) , with R X δ () d P = 1, gives rise to a new probability measure μ (the text calls it P ) by μ ( A ) := Z A δ () d P . Prove, for each measurable B , that μ ( T-n ( B )) → P ( B ) as n → ∞ . [ Hint: You might ﬁrst want to consider the case where δ is a ( scaled ) indicator function [1 / P ( A )] 1 A .]...
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