restriction of sigma-algebra

**Dinkydoe** · October 15th 2011, 04:37 AM

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.

Thread: restriction of sigma-algebra

LinkBack

Thread Tools

Display

restriction of sigma-algebra

Similar Math Help Forum Discussions

Sigma-algebra

Is a countable union/intersection of sigma-algebra also a sigma-algebra?

Show intersection of sigma-algebras is again a sigma-algebra

sigma algebra over R

Sigma-Algebra

Search Tags