Remark

— 1) For a linear mapping m of $\mathscr{K}(T)$ into F to be a vectorial measure, it is necessary and sufficient that, for every compact subset K of T, the image under m of the unit ball $\|f\| \leq 1$ of $\mathscr{K}(T, K)$ be bounded in F. The notion of vectorial measure with values in F is therefore the same for all the Hausdorff locally convex topologies on F that admit the same bounded sets, and in particular for all the topologies compatible with the duality between F and F' (TVS, IV, §1, No. 1, Prop. 1).

2) Let $T_1$ be a locally compact space, $F_1$ a Hausdorff locally convex space over $\mathbf{R}$, u a continuous linear mapping of $\mathscr{K}(T_1)$ into $\mathscr{K}(T)$, and v a continuous linear mapping of F into $F_1$. If m is a vectorial measure on T with values in F, then $v \circ m \circ u$ is a vectorial measure on $T_1$ with values in $F_1$. In particular, if g is a continuous, finite numerical function on T, then $f \mapsto m(gf)$ is a vectorial measure on T with values in F, which is denoted $g \cdot m$; if h is a second continuous, finite numerical function on T, then $g \cdot (h \cdot m) = (gh) \cdot m$.

3) Since the space $\mathscr{K}(T)$ is the direct limit of the Banach spaces $\mathscr{K}(T, K)$, and is in particular barreled (TVS, III, §4, No. 1, Cor. of Prop. 2 and Cor. 3 of Prop. 3), in order that a linear mapping m of $\mathscr{K}(T)$ into F be a vectorial measure, it is necessary and sufficient that, for every $z' \in F'$, $z' \circ m$ be a scalar measure on T (TVS, II, §6, No. 4, Prop. 5 and IV, §1, No. 3, Prop. 7).

4) In view of Remark 1, Prop. 1 of Ch. III, §2, No. 1 and its proof are again valid for vectorial measures. One can therefore define the support of a vectorial measure m on T to be the complement of the largest open set $U \subset T$ such that the restriction of m to U is zero.