Proposition 7

— *Let T be a subspace of a topological space X, and let i be the injection of T into X.

a) *If $\mu$ is a measure on T, and if i is $\mu$-proper, then the measure $i(\mu)$ is concentrated on T, and one has $(i(\mu))_T = \mu$.

b) *If $\lambda$ is a measure on X such that T is $\lambda$-measurable, then i is $\lambda_T$-proper, and $i(\lambda_T) = \varphi_T \cdot \lambda$.

a) Set $\nu = i(\mu)$; the relation $\nu^\bullet(A) = \mu^\bullet(A \cap T)$, applied to $A = X - T$, shows that $\nu$ is concentrated on T. The relation $\nu_T = \mu$ is a special case of the relation (2), on taking $B = T = A$.

b) Let $f$ be a positive function defined on X; setting $\mu = \lambda_T$, one has $\mu^\bullet(f \circ i) = \lambda_T^\bullet(f_T) = \lambda^\bullet(f \varphi_T) \leq \lambda^\bullet(f)$ (Prop. 1); it follows that i is $\mu$-proper. On the other hand, $\mu^\bullet(f \circ i)$ (resp. $\lambda^\bullet(f \varphi_T)$) is the essential upper integral of $f$ with respect to $i(\mu)$ (resp. $\varphi_T \cdot \lambda$). These two measures are therefore equal.

Remark 6). — Let $\pi$ be a $\mu$-proper mapping of $T$ into a topological space $X$. One reduces the theory of integration with respect to the image measure $\nu = \pi(\mu)$ to the theory treated in Ch. V, §6, in the following way. Let $(K_\alpha)_{\alpha \in A}$ (resp. $(L_\beta)_{\beta \in B}$) be a crushing of $T$ (resp. of $X$) for $\mu$ (resp. for $\nu$), and set $N = T - \bigcup_{\alpha \in A} K_\alpha,\ P = X - \bigcup_{\beta \in B} L_\beta$. We can suppose that the restriction of $\pi$ to each $K_\alpha$ is continuous (§1, No. 8, Prop. 10). Let $T', X'$ be the locally compact spaces constructed as in the Scholium of §1, No. 8 and let $\mu'$ and $\nu'$ be the measures on these spaces associated with $\mu$ and $\nu$. The topology of $T'$ being the sum of the topologies of the subspaces $K_\alpha$ and the discrete topology on $N$, $\pi$ is a continuous mapping of $T'$ into $X$ and the relation ${\mu'}^\bullet(g \circ \pi) = \mu^\bullet(g \circ \pi) = \nu^\bullet(g)$ (for $g \in \mathscr{F}_+(X)$) shows that $\pi$ is $\mu'$-proper and that $\pi(\mu') = \nu$. On the other hand, the identity mapping $i$ of $X$ onto $X'$ is $\nu$-proper, and $i(\nu) = \nu'$. It follows that $\pi$ is a $\mu'$-proper mapping of $T'$ into $X'$, and that the image of $\mu'$ under $\pi$ is $\nu'$ (Cor. of Prop. 5). We leave to the reader the task of transcribing the results of Ch. V, §6.