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Math Help - restriction of sigma-algebra

  1. #1
    Senior Member Dinkydoe's Avatar
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    restriction of sigma-algebra

    I hope some of you can help me out a little.

    I read the following in a article, like it's an obvious fact. (http://research.microsoft.com/en-us/...papers/poi.pdf page 18, lemma 11)

    Let \mathcal{F} be a \sigma-algebra on the probability space \Omega=\mathbb{R}, and let
    S=[-r,r] be some bounded interval.

    Let \mathcal{F}_S be the \sigma-algebra, generated by the restriction of \mathcal{F} to S. Let A\in \mathcal{F} = \mathcal{F}_{\mathbb{R}}. Why does there exist for every  \epsilon>0 some r=r(\epsilon) and an event A_{\epsilon}\in \mathcal{F}_{[-r,r]} such that \mathbb{P}(A\Delta A_{\epsilon}) < \epsilon. ??

    In otherwords, A can sufficiently be approximated by A_{\epsilon}. Is this obvious?
    Is it because \lim_{r\to\infty}\mathcal{F}_{[-r,r]}=\mathcal{F}...(and continuity of the probability measure \mathbb{P}?)

    Also is claimed that \mathcal{F}_{(-2r,0]}\subset \mathcal{F}_{(-\infty,0]} Is this true?

    Something general like \mathcal{F}_A\subset \mathcal{F}_B if A\subset B, does not hold right?

    Really looking forward to your suggestions.
    Last edited by Dinkydoe; October 15th 2011 at 08:39 AM.
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