Exercise 23, § 5

Let $\mu$ be a positive measure on a locally compact space $T$, $p$ a number such that $1 \leq p < +\infty$, and $q$ the conjugate exponent. Show that if $g$ is a finite measurable function such that, for every function $f \in \mathscr{L}^p$, the function $fg$ is integrable, then $g$ is locally almost everywhere equal to a function in $\mathscr{L}^q$. (Show that the mapping $f \mapsto fg$ of $\mathscr{L}^p$ into $\mathscr{L}^1$ is continuous, using the closed graph theorem (TVS, I, §3, Cor. 5 of Th. 1).)