Proposition 7

*Let $\theta$ be a complex measure on $T$, $\pi$ a $\theta$-proper mapping of $T$ into a locally compact space $X$, $\nu$ the image measure $\pi(\theta)$.

a) Let $A$ be a subset of $X$; if $\overline{\pi^{-1}}(A)$ is locally $\theta$-negligible, then $A$ is locally $\nu$-negligible.

b) Let $f$ be a mapping of $X$ into a topological space; if $f \circ \pi$ is $\theta$-measurable, then $f$ is $\nu$-measurable.*

c) Let $f$ be a function defined on $X$, with values in a Banach space $F$; if $f \circ \pi$ is essentially $\theta$-integrable, then $f$ is essentially $\nu$-integrable and

Taking into account formula (8), these results may be deduced from Cor. 2 of Prop. 2, from Prop. 3, and from Th. 1 of No. 2.