Remark

— 1) Let $\mathscr{E}$ be a dense linear subspace of $\mathscr{L}_{F'}^q$; then the formula (9) holds when $g$ runs over the intersection of $\mathscr{E}$ with the set $B$ of functions in $\mathscr{L}_{F'}^q$ such that $N_q(g) \leq 1$. For, it suffices to observe that the interior $\overset{\circ}{B}$ of $B$ is dense in $B$ and that $\overset{\circ}{B} \cap \mathscr{E}$ is dense in $\overset{\circ}{B}$, since $\overset{\circ}{B}$ is open. This remark applies in particular to the set $\mathscr{E} = \mathscr{K}(X; F')$ of continuous functions with compact support (with values in $F'$) when $1 \leq q < +\infty$, that is, $1 < p \leq +\infty$. But in this case, the formula (9) is true as $g$ runs over $B \cap \mathscr{K}(X; F')$, even for $p = 1$. For, we may, as above, restrict ourselves to the case that $f$ is a step function. We saw then that if $N_1(f) = 1$, then for every $\varepsilon > 0$ there exists a step function $g \in \mathscr{L}_{F'}^\infty$ such that $|g(x)| \leq 1$ for all $x \in X$ and $|\int \langle f, g \rangle d\mu| \geq 1 - \varepsilon$. There exists a finite number of pairwise disjoint compact sets $K_i$ such that $g$ has a constant value $a'_i$ on each $K_i$ and such that, if $K$ is the union of the $K_i$, then $\int |f| \varphi_{\mathscr{C}_K} d\mu \leq \varepsilon$. Let $U_i$ be a neighborhood of $K_i$ such that the sets $U_i$ are pairwise disjoint, and let $h_i$ be a continuous mapping of $X$ into $[0, 1]$ with support contained in $U_i$ and equal to 1 on $K_i$. Setting $h = \sum a'_i h_i$, we have $h(x) = g(x)$ on $K$ and $|h(x)| \leq 1$ on $X$, therefore

and consequently $|\int \langle f, h \rangle d\mu| \geq 1 - 3\varepsilon$, which proves our assertion. Analogous remarks can be made for the formula (10).

2) Let $\mu$ be a positive measure on $X$, $f$ a measurable function $\geq 0$ (finite or not) whose support is contained in a countable union of compact sets $K_n$. Then, for every $p$ such that $1 \leq p \leq +\infty$,

as $g$ runs over the set of functions in $\mathscr{H}(X; \mathbf{R})$ such that $N_q(g) \leq 1$. For, the formula (11) is a special case of (9) when $N_p(f) < +\infty$, since $f$ is then equivalent to a function in $\mathscr{L}^p$ (§ 5, No. 6, Th. 5). If $N_p(f) = +\infty$, for every integer $n > 0$ set $f_n = \inf(n, f \varphi_{K_n})$. Then

whence, on passing to the limit (assuming, as we may, that the sequence $(K_n)$ is increasing), we have $\sup \int^* |f g| d\mu = +\infty$ (§ 1, No. 3, Th. 3).