Proposition 20

— Let F be a Banach space, equipped with the uniform structure defined by its norm.

(i) For every $\mu$-measurable subset A of X, the topology of convergence in measure is compatible with the vector space structure of $\mathscr{S}(A, \mu; F)$.

(ii) The space $\mathscr{K}(X; F)$ is dense in $\mathscr{S}(X, \mu; F)$.

(iii) For every real number $p \geq 1$, the topology induced on the space $\mathscr{L}_F^p(X, \mu)$ by the topology of convergence in measure is coarser than the topology of convergence in mean of order $p$.

(i) For every $\mu$-integrable subset B of A and every $\delta > 0$, denote by $T(B, \delta)$ the set of $f \in \mathscr{S}(A, \mu; F)$ for which the set C of $x \in B$ such that $|f(x)| \geq \delta$ satisfies the relation $|\mu|(C) \leq \delta$; it is clear that if $V_\delta$ is the entourage of F formed by the pairs $(y, z)$ such that $|y - z| \leq \delta$, the entourage $W(V_\delta, B, \delta)$ is the set of pairs $(f, g)$ of measurable mappings of A into F such that $f - g \in T(B, \delta)$. It is clear that the sets $T(B, \delta)$ are symmetric, and that $T(B, \delta) + T(B, \delta) \subset T(B, 2\delta)$ and $T(B, |\alpha|\delta) \subset \alpha T(B, \delta)$ for every nonzero scalar $\alpha$ such that $|\alpha| \leq 1$; it therefore suffices to verify that the sets $T(B, \delta)$ are absorbent (TVS, I, §1, No. 5, Prop. 4). Now, if $f$ is a $\mu$-measurable mapping of A into F, the numerical function $|f|$ is also $\mu$-measurable (No. 3, Cor. 6 of Th. 1). Let $C_n$ be the set of $x \in B$ such that $|f(x)| \geq n$; the $C_n$ form a decreasing sequence of integrable sets whose intersection is empty; therefore there exists an integer $n$ such that $|\mu|(C_n) \leq \delta$ (§ 4, No. 5, Cor. of Prop. 7); we can moreover suppose that $n$ is taken sufficiently large that $1/n \leq \delta$; then $f/n^2 \in T(B, \delta)$, which completes the proof of assertion (i).

(iii) The relation $\int |f|^p d|\mu| \leq \delta^{p+1}$ implies that if $C$ is the set of $x \in X$ such that $|f(x)| \geq \delta$, then

whence $|\mu|^*(C) \leq \delta$, which proves (iii).

(ii) In view of (iii), it suffices to show for example that $\mathscr{L}_F^1$ is dense in $\mathscr{S}_F$, since by definition $\mathscr{H}(X; F)$ is dense in $\mathscr{L}_F^1$ for the topology of convergence in mean. Now, let $f$ be any element of $\mathscr{S}_F$ and let $T(B, \delta)$ be a neighborhood of 0 in this space; we see as in (i) that there exists an integrable subset $C$ of $B$ such that $|\mu|(C) \leq \delta$ and such that $f$ is bounded on $B - C$; denoting then by $g$ the function equal to $f$ on $B - C$ and to 0 on $X - (B - C)$, it follows from No. 6, Th. 5 that $g$ is integrable, and obviously $f - g \in T(B, \delta)$.