Scholium

— Let $(K_\alpha)_{\alpha \in A}$ be a crushing of T, and let $N = T - \bigcup_\alpha K_\alpha$. We denote by $T'$ the locally compact space obtained by equipping $T'$ with the sum topology of the topologies of the subspaces $K_\alpha$ and any locally compact topology on $N$ (unless expressly mentioned to the contrary, $N$ will always be equipped with the discrete topology). For each $\alpha \in A$, let $i_\alpha$ be the canonical injection of $K_\alpha$ into $T'$, and let $\mu'_\alpha$ be the measure on $T'$ that is the image of $\mu_{K_\alpha}$ under $i_\alpha$. The family $(\mu'_\alpha)$ is summable: for, if $f$ is a continuous function on $T'$ with compact support, then $\operatorname{Supp}(f)$ intersects $K_\alpha$ for only a finite number of indices $\alpha$. We set $\mu' = \sum_{\alpha \in A} \mu'_\alpha$. The set $N$ being locally negligible for $\mu'$, since it is so for each $\mu'_\alpha$ (Prop. 9), the family $(K_\alpha)_{\alpha \in A}$ is a $\mu'$-crushing of $T'$; now, the measure induced by $\mu'$ on $K_\alpha$ is obviously $\mu_{K_\alpha}$ and the formula (7), applied to $\mu$ and to $\mu'$, shows that $\mu^\bullet = {\mu'}^\bullet$. Similarly, the last assertion of the statement of Prop. 9, applied to $\mu$ and to $\mu'$, shows that the measurable mappings are the same for the two measures $\mu$ and $\mu'$.

These two properties permit reducing nearly all of the theory of integration with respect to $\mu$ to the theory elaborated for locally compact spaces. These considerations will be developed in No. 10.

Here is another application of the concept of crushing: