Suppose $1 \leq p < \infty$, and $q$ is the expoent conjugate to $p$. Suppose that $\mu$ is a $\sigma-$finite measure and $g$ is a measurable function such that $fg \in L^1(\mu)$ for every $f\in L^p(\mu)$. Prove that $g \in L^q(\mu)$.
What i did : For $1 \leq p < \infty$ we can define a linear functional $\Phi : L^p(\mu) \rightarrow \mathbb{R}$ give by $\Phi(f) =\displaystyle \int_{X}fg d\mu$; by the characterization of linear functionals on $L^p(\mu)$ $g \in L^q(\mu)$. (This is right?)
But i don't know as procced for the case $p = \infty$. Any tip would great.