Definition 1

Let $\mu$ be a positive measure on a locally compact space $T$. A mapping $\pi$ of $T$ into a locally compact space $X$ is said to be $\mu$*-proper* (or proper for the measure $\mu$) if the pair $(\pi, 1)$ is $\mu$*-adapted*, that is (§ 4, No. 1), if $\pi$ is $\mu$*-measurable and satisfies the (equivalent) conditions of Prop. 1*. The measure $\int \varepsilon_{\pi(t)} \, d\mu(t)$ on $X$ is then called the image of $\mu$ under $\pi$ and is denoted $\pi(\mu)$.

Thus if $\nu = \pi(\mu)$ then, by definition, for $f \in \mathscr{K}(X)$ one has