If $\displaystyle (X,\Omega,\mu) $ is a $\displaystyle \sigma$-finite measure space, $\displaystyle \phi:X \rightarrow F$ is an $\displaystyle \Omega$-measureable function, $\displaystyle 1\leq p \leq \infty$, and $\displaystyle \phi f \in L^{p}(\mu)$, then show that $\displaystyle \phi \in L^{\infty}(\mu)$.

Any ideas?