Proposition 14

— In order that a positive linear form $\theta$ on $\mathscr{L}^\infty$ be of the type $f \mapsto \int f g d\mu$, where $g \in \mathscr{L}^1$, it is necessary and sufficient that, for every increasing directed family $(f_\alpha)_{\alpha \in A}$ of positive functions in $\mathscr{L}^\infty$ whose set of classes $(\dot{f}_\alpha)_{\alpha \in A}$ is bounded above in $L^\infty$ and admits $\dot{h}$ as its supremum in this space, one have

Let us first show that the condition is necessary. The measure $h \cdot \mu$ is the supremum in $\mathscr{M}(T)$ of the set of measures $f_\alpha \cdot \mu$ (No. 5, Scholium); therefore (Ch. II, §2, No. 2), for every function $\varphi \geq 0$ in $\mathscr{K}(T)$, we have $\int h \varphi \, d\mu = \sup_{\alpha \in A} \int f_\alpha \varphi \, d\mu$. If now $a$ is a number $\geq 0$ such that $N_\infty(f_\alpha) \leq a$ for all $\alpha \in A$ (which implies that $N_\infty(h) \leq a$), then for every $\varepsilon > 0$ there exists a $\varphi \in \mathscr{K}(T)$ such that $\varphi \geq 0$ and $N_1(g - \varphi) \leq \varepsilon$, from which we infer that $\int f_\alpha |g - \varphi| \, d\mu \leq a \varepsilon$ for all $\alpha \in A$, and $\int h |g - \varphi| \, d\mu \leq a \varepsilon$. Since $\sup_{\alpha \in A} \int f_\alpha g \, d\mu \leq \int h g \, d\mu$, this proves that the two members of this inequality are equal.

To establish that the condition is sufficient, we shall make use of the following lemma: