Exercise 14, § 5

Let $\theta$ be a linear form on the space $\mathscr{L}^\infty(T, \mu)$; in order that $\theta$ be of the type $f \mapsto \int fg \, d\mu$, where $g \in \mathscr{L}^1(T, \mu)$, it is necessary and sufficient that $\theta$ satisfy the following conditions: 1° for every $\varepsilon > 0$, there exists a $\delta > 0$ such that the relation $\mu^*(A) \leq \delta$ implies $|\theta(h)| \leq \varepsilon$ for every $\mu$-measurable function $h$ such that $|h| \leq \varphi_A$; 2° for every $\varepsilon > 0$, there exists a compact set K such that, for every $\mu$-integrable set $B \subset T - K$, one has $|\theta(\varphi_B)| \leq \varepsilon$. (Use the Lebesgue–Nikodym theorem.)