Definition 2

Let $m$ be a vectorial measure on $T$ with values in $F$, $f$ a numerical function on $T$ that is essentially integrable for $m$. One calls integral of $f$ with respect to $m$, and denotes by $m(f)$ or $\int f \, dm$ or again $\int f(t) \, dm(t)$, the element of ${F'}^*$ defined by

We observe that if $f \in \mathscr{K}(T)$, the element $\int f \, dm$ so defined coincides with the element denoted likewise in No. 1, because the second member of (1) is then $\int f \, d(z' \circ m) = z'(m(f))$ by definition. Moreover, if in particular one applies Def. 2 to the case that $F = \mathbf{R}$, one sees that for every $z' \in F'$, f is essentially integrable for the real measure $z' \circ m$, and that the second member of (1) may be written $\int f d(z' \circ m)$.

Suppose now that $f$ is essentially integrable for $m$, and let $z' \in F'$. Set $\mu = |z' \circ m|$; by the Lebesgue–Nikodym theorem, one can write $z' \circ m = g \cdot \mu$, where $g$ is locally $\mu$-integrable and $\|g\| \leq 1$, and the proof of Prop. 1 shows that $\int f d(z' \circ m) = \int fg d\mu$. Consequently,

It is clear that the set of finite numerical functions essentially integrable for $m$ is a vector space over $\mathbf{R}$; we shall denote by $\mathscr{L}(m)$ this space equipped with the coarsest locally convex topology making continuous all of the linear forms $f \mapsto \int f d(z' \circ m)$, where $z'$ runs over $F'$. Note that in general the locally convex space $\mathscr{L}(m)$ is not Hausdorff.