Definition 2

— A continuous linear form on $\mathscr{K}(X; \mathbf{C})$, X a locally compact space, is called a measure (or complex measure) on X.

If $\mu$ is a measure on a locally compact space $X$, the value of the measure for a function $f \in \mathscr{K}(X; \mathbf{C})$ is called the integral of f with respect to $\mu$; besides the general notations $\mu(f)$ and $\langle f, \mu \rangle$, one also uses the notations $\int f d\mu$, $\int f \mu$, $\int f(x) d\mu(x)$ and $\int f(x) \mu(x)$ to denote it; as to the use of the letter $x$, see S, I, §1, No. 1.

By virtue of the criterion for continuity in direct limits (TVS, Ch. II, §4, No. 4, Prop. 5), to say that $\mu$ is a measure on $X$ means that $\mu$ is a linear form on $\mathscr{K}(X; \mathbf{C})$ satisfying the following condition: for every compact subset $K$ of $X$, there exists a number $M_K$ such that, for every function $f \in \mathscr{K}(X; \mathbf{C})$ whose support is contained in $K$,

More generally: