Definition 2

Let $\theta$ be a complex measure on a locally compact space $T$. A mapping $\pi$ of $T$ into a locally compact space $X$ is said to be $\theta$-proper if it is $|\theta|$-proper. The measure $f \mapsto \int (f \circ \pi) \, d\theta$ is then called the image of $\theta$ under $\pi$ and is denoted $\pi(\theta)$..

The relation (7) may then be put in the following form:

The measure $\pi(\theta)$ may be zero without $\theta$ being zero, as one sees immediately on taking for $T$ a space reduced to two points $a, b$, for $\theta$ the measure $\varepsilon_a - \varepsilon_b$, and for $\pi$ a constant mapping.

Let $\theta$ and $\theta'$ be two complex measures on $T$; if $\pi$ is $\theta$-proper and $\theta'$-proper, it follows from Cor. 2 of Prop. 6 that $\pi$ is $(\theta + \theta')$-proper since $|\theta + \theta'| \leq |\theta| + |\theta'|$, and obviously $\pi(\theta + \theta') = \pi(\theta) + \pi(\theta')$.

In particular, if $\theta$ is a real measure and $\pi$ is $\theta$-proper, then

Several results established earlier extend at once to complex measures; we cite the most important among them.