Exercise 6, § 1

Let $\mathscr{K}$ be a $\mu$-dense set of compact subsets of T. Let $f$ be a function defined on T, with values in a Banach space F, such that $f \varphi_K$ is $\mu$-integrable for every $K \in \mathscr{K}$. If the integrals $\int f \varphi_K \, d\mu$ have a limit in F with respect to the directed set $\mathscr{K}$ (for $\subset$), then $f$ is essentially $\mu$-integrable.