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 Post subject: LimitPosted: Mon, 1 Aug 2011 13:17:39 UTC
 S.O.S. Oldtimer

Joined: Thu, 30 Oct 2008 10:47:09 UTC
Posts: 182
Suppose f is absolutely continuous on , and . Prove that
a) exists and finite for ,
b) , for .

How to relate the limit to given integral?

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 Post subject: Re: LimitPosted: Mon, 1 Aug 2011 19:51:31 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
What is p in this case? Is it for all p, or just some p?

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 Post subject: Re: LimitPosted: Mon, 1 Aug 2011 23:03:05 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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Beta wrote:
Suppose f is absolutely continuous on , and . Prove that
a) exists and finite for ,
b) , for .

How to relate the limit to given integral?

I think you want , rather than only some , that is absolutely continuous.

If p>2, then from measurability of f' and integrability of , you know that is (uniformly) bounded for all (OK, technically speaking you want a representative of f' so that blahblahblah). Thus by choosing small enough you can make sure and hence with q<1, so dominated convergence gives...

For p=2, use a similar idea but dominate with instead of .

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