Corollary 3

— For every real measure $\mu$ on a locally compact space $X$,

It suffices to make use of the formula (22) and the expression (9) for $|\mu|(f)$ when $\mu$ is a real measure and $f \in \mathscr{K}_+(X)$.

The set of bounded real measures is therefore the dual of the normed space $\mathscr{K}(X; \mathbf{R})$; it is denoted $\mathscr{M}^1(X, \mathbf{R})$, or $\mathscr{M}^1(X)$ if no confusion can result. The canonical injection $\mathscr{M}^1(X, \mathbf{R}) \to \mathscr{M}^1(X; \mathbf{C})$ is an isometry by virtue of (24).