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 be a -algebra on the probability space , and let

be some bounded interval.

Let be the -algebra, generated by the restriction of to . Let . Why does there exist for every some and an event such that . ??

In otherwords, can sufficiently be approximated by . Is this obvious?

Is it because ...(and continuity of the probability measure ?)

Also is claimed that Is this true?

Something general like if , does not hold right?

Really looking forward to your suggestions.