Remark 2

It is clear that the integral of f depends only on the class of f modulo the space of mappings of T into F that are scalarly locally $\mu$-negligible. Note that a scalarly locally negligible function g is not necessarily zero almost everywhere (Exer. 12). However, this is indeed the case when there exists in F' a sequence $(\mathbf{z}'_n)$ that is dense for the topology $\sigma(F', F)$: for, if $H_n$ is the locally negligible set of points $t \in T$ such that $\langle g(t), \mathbf{z}'_n \rangle \neq 0$, the union H of the $H_n$ is locally negligible and, for every $t \notin H$, one has $\langle g(t), \mathbf{z}'_n \rangle = 0$ for all n, whence $g(t) = 0$.

Let u be a continuous linear mapping of F into a Hausdorff locally convex space G; its transpose $^t u$ is a linear mapping of $G'$ into $F'$, and the (algebraic) transpose $^t(^t u)$ is a linear mapping of ${F'}^*$ into ${G'}^*$ that extends u, which we shall again denote by u. With this convention: