Definition 1

Let F be a Hausdorff locally convex space over R. One calls vectorial measure on T with values in F every continuous linear mapping of the space $\mathscr{K}(T)$ into F.

Def. 1 may also be expressed as follows: a vectorial measure on T with values in F is a linear mapping m of $\mathscr{K}(T)$ into F such that, for every compact subset K of T, the restriction of m to $\mathscr{K}(T, K)$ is continuous for the topology of uniform convergence. If $f \in \mathscr{K}(T)$, one also writes $\int f dm$ or $\int f(t)\,dm(t)$ instead of $m(f)$. The measures with values in $\mathbf{R}$ are sometimes called real measures (Ch. III, §1, No. 5) or scalar measures on T.