Suppose f is absolutely continuous on
. Prove that
exists and finite for
How to relate the limit to given integral?
I think you want
, rather than only some
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
with q<1, so dominated convergence gives...
For p=2, use a similar idea but dominate with