Corollary 2

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

(23)

This formula will be generalized in Ch. IV, §4, No. 7.

On a compact space $X$, for every (complex) measure $\mu$ on $X$ the complex number $\mu(1)$ is called the total mass of $\mu$. When $\mu$ is positive, its total mass is thus equal to its norm. When $\mu$ is a positive measure on a compact space $X$, of total mass equal to 1, one also says that its value $\mu(f)$ for a continuous function $f \in \mathscr{C}(X; \mathbf{C})$ is the mean of $f$ with respect to the measure $\mu$.