Hmm. I seem to remember there are theorems about for certain conditions on and . Look in Royden under that subject.
A real-valued measurable function f is said to be in -space if has finite integral. So if , must be Lebesgue integrable. My question is: in this case, is it possible that f itself is not Lebesgue integrable? Under what condition can we have that an f in is also Lebesgue integrable? Thanks!