The question is simply: If I have two random variables $X,Y \in \bigcap_{p\geq 1} L^p(\Omega)$ and a third $Z$ r.v. such that $$XZ=Y$$ can I conclude that $Z\in \bigcap_{p\geq 1}L^p(\Omega)$? (I know this would't hold for a single $L^p(\Omega)$ space, the point is that both $X$ and $Y$ are in all $L^p(\Omega)$, $p\geq 1$)
Or, in other words, if $X\neq 0$, $X\in \bigcap_{p\geq 1} L^p(\Omega)$ is then $1/X \in \bigcap_{p\geq 1} L^p(\Omega)$?
Thank you very much for the help!
Take $\Omega:=(0,1)$ with Lebesgue measure and $X(\omega):=\omega$. $X$ is bounded, hence in all the $L^p$ (as $\Omega$ has finite measure) but its reciprocal is not even integrable.