1. A problem on L^p-space

A real-valued measurable function f is said to be in $L^p$-space if $|f|^p$ has finite integral. So if $f\in L^p$, $|f|^p$ 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 $L^p$ is also Lebesgue integrable? Thanks!

2. Hmm. I seem to remember there are theorems about $L^{p}\subset L^{q}$ for certain conditions on $p$ and $q$. Look in Royden under that subject.

3. I searched chapter 6 in Royden's book, as well as exercises, but failed to find such theorem. Is it located in some other chapter?

4. Originally Posted by zzzhhh
A real-valued measurable function f is said to be in $L^p$-space if $|f|^p$ has finite integral. So if $f\in L^p$, $|f|^p$ 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 $L^p$ is also Lebesgue integrable? Thanks!
You can take a trivial counter-example:
$
\int |\frac{1}{x} \mathbf{1}_{[1, \infty)}|= \infty
$

Now take p=2 for a contrast.

As for $L^p \subset L^1$, this is true under finite measures (you can use Jensen). I don't know about conditions on general spaces but being compactly supported is obviously enough.