Example

— Let $X$ and $T$ be two locally compact spaces, and let $\pi$ be a local homeomorphism of $X$ into $T$. In other words (GT, I, §11, Exer. 25), we assume that every point $x \in X$ admits a neighborhood $V$ such that $\pi|V$ is a homeomorphism of $V$ onto a neighborhood of $\pi(x)$; if necessary replacing $V$ by a relatively compact open neighborhood $W$ of $x$ such that $\overline{W} \subset V$, one deduces that the set $\mathscr{U}$ of relatively compact open subsets $U$ of $X$, such that $\pi|\overline{U}$ is a homeomorphism of $\overline{U}$ onto its image, is an open covering of $X$. Now let $\mu$ be a positive measure on $T$; if $U$ is an element of $\mathscr{U}$, then $\pi(U)$ is an open set in the compact space $\pi(\overline{U})$, therefore is a locally compact subspace of $T$, and one knows how to define the measure $\mu|\pi(U)$ induced by $\mu$ on $\pi(U)$ (Ch. IV, §5, No. 7). Let $\nu_U$ be the image of $\mu|\pi(U)$ under the homeomorphism inverse to $\pi|U$; we are going to show that there exists one and only one measure $\nu$ on $X$ that induces the measure $\nu_U$ on every open set $U \in \mathscr{U}$. This measure is called the inverse image of $\mu$ under the local homeomorphism $\pi$, and is denoted $\pi^{-1}(\mu)$.

The uniqueness of $\nu$ follows at once from the principle of localization (Ch. III, §2, No. 1, Cor. of Prop. 1). To establish existence, we note that if $t \in T$, then every point $x \in \pi^{-1}(t)$ admits a neighborhood that intersects $\pi^{-1}(t)$ only at the point $x$, so that $\pi^{-1}(t)$ is a discrete subspace of $X$, and that the family $(\varepsilon_x)_{x \in \pi^{-1}(t)}$ is summable; we denote its sum by $\lambda_t$. We next show that the mapping $t \mapsto \lambda_t$ is scalarly essentially $\mu$-integrable, and that its integral $\nu = \int \lambda_t d\mu(t)$ is the sought-for inverse image. This will result at once from the following lemma: