Suppose $X,Y$ r.v's on a probability space, such that $E|X|^p<\infty$ and $E|Y|^q<\infty$, for $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$. Is it true that $E|XY|<\infty$?
My question comes from this: I have to prove the conditional Holder's inequality under these assumptions. And I was wondering whether the LHS of it, namely $E[|XY|\, \, | \mathcal{G}]$ is at all defined and I couldn't come up with a proof that $XY$ is indeed integrable.