Chapter VI

Vectorial integration

In this chapter, if F denotes a Hausdorff locally convex vector space (over $\mathbf{R}$ or $\mathbf{C}$), we denote by $F'$ its dual, by $F''$ its bidual, and by ${F'}^*$ the algebraic dual of $F'$ (the space of all linear forms on $F'$); $F''$ is a linear subspace of ${F'}^*$, and F may be identified (as a vector space without topology) with a linear subspace of $F''$. We denote by $F_\sigma$ the vector space F equipped with the weakened topology $\sigma(F, F')$; the qualifiers 'weak' and 'weakly' refer to this topology.

In this chapter, T denotes a locally compact space, $\mathscr{K}_\mathbf{R}(T)$ or $\mathscr{K}(T)$ (resp. $\mathscr{K}_\mathbf{C}(T)$) the vector space of real (resp. complex) functions on T, continuous and with compact support; for every subset A of T, $\mathscr{K}(T, A)$ (resp. $\mathscr{K}_\mathbf{C}(T, A)$) denotes the subspace of $\mathscr{K}(T)$ (resp. $\mathscr{K}_\mathbf{C}(T)$) formed by the functions whose support is contained in A. Absent express mention to the contrary, the space $\mathscr{K}(T)$ (resp. $\mathscr{K}_\mathbf{C}(T)$) will be equipped with the direct limit topology of the topologies of uniform convergence on each of the subspaces $\mathscr{K}(T, K)$ (resp. $\mathscr{K}_\mathbf{C}(T, K)$), where K runs over the set of compact subsets of T.

We recall that this topology is finer than the topology of uniform convergence, hence is Hausdorff; it induces on each $\mathscr{K}(T, K)$ (resp. $\mathscr{K}_\mathbf{C}(T, K)$) the topology of uniform convergence (TVS, II, §4, No. 4, Remark). The space $\mathscr{K}_\mathbf{C}(T)$ may be identified with the space obtained from $\mathscr{K}(T)$ by extension of the scalars from $\mathbf{R}$ to $\mathbf{C}$. To say that a linear form on $\mathscr{K}(T)$ is a measure means, by definition, that it is continuous (Ch. III, §1, No. 3, Def. 2).