Proposition 12

— For a measure $\mu$ on a locally compact space $X$ to be bounded (Ch. III, §1, No. 8), it is necessary and sufficient that $X$ be an integrable set with respect to $\mu$ (or, what comes to the same thing, that every finite constant function be integrable); in this case,

For, we have seen that $|\mu|^*(X) = \| \mu \|$ (§ 1, No. 2); the proposition therefore follows from Prop. 10 of No. 6.

For every bounded measure $\mu$, we again say that $\mu(X)$ is the total mass of $\mu$.

It follows from Th. 4 of No. 6 that if $\mu$ is a bounded measure then, for every $\varepsilon > 0$, there exists a compact set $K$ such that $|\mu|(CK) \leq \varepsilon$.