Theorem 2

— *Let T and B be two locally compact spaces having countable bases, $\mu$ a positive measure on T, $p$ a $\mu$-measurable mapping of T into B, and $\nu$ a pseudo-image measure of $\mu$ under $p$. Then there exists a $\nu$-adequate family $b \mapsto \lambda_b$ ($b \in B$) of positive measures on T, having the following properties: a) $\lambda_b \neq 0$ for $b \in p(T)$; b) $\lambda_b$ is concentrated on the set $\overline{p^{-1}}(b)$ for every $b \in B$; in particular, $\lambda_b = 0$ for $b \notin p(T)$; c) $\mu = \int \lambda_b \, d\nu(b)$.

Moreover, if $\nu' = r \cdot \nu$ is a second pseudo-image measure of $\mu$ under $p$, and if $b \mapsto \lambda'_b$ is a $\nu'$-adequate family of positive measures on T having the properties b) and c) with respect to $\nu'$, then $\lambda_b = r(b)\lambda'_b$ almost everywhere in B (for $\nu$ or $\nu'$).

There exists a continuous and finite numerical function $f$ defined on T, such that $f(t) > 0$ for every $t \in T$ and such that $\mu'' = f \cdot \mu$ is bounded (Ch. V, §5, No. 6, Prop. 11). Let $\nu'' = p(\mu'')$, which is equivalent to $\nu$, and write $\nu'' = g \cdot \nu$, with $g$ finite and locally $\nu$-integrable; one can suppose, moreover, that $g(b) > 0$ for all $b \in B$ (Ch. V, §5, No. 6, Prop. 10). Th. 1 of No. 1, applied to $\mu''$ and $\nu''$, shows that there exists a $\nu''$-adequate family $b \mapsto \lambda''_b$ ($b \in B$) of positive measures on T, such that:

1) $\| \lambda''_b \| = 1$ for $b \in p(T)$; 2) $\lambda''_b$ is concentrated on $\overline{p^{-1}}(b)$ for every $b \in B$; 3) $\mu'' = \int \lambda''_b \, d\nu''(b)$.

For every $b \in B$, let us define a positive measure $\lambda_b$ on T by the formula $\lambda_b = (1/f) \cdot (g(b)\lambda''_b)$. It is clear that the family $b \mapsto \lambda_b$ has the properties a) and b) of the statement. On the other hand, for every function $h \in \mathscr{K}(T)$, $h/f$ belongs to $\mathscr{K}(T)$, therefore

But since the function $b \mapsto \int (h(t)/f(t)) \, d\lambda''_b(t)$ is $\nu''$-integrable, the function $b \mapsto g(b) \int (h(t)/f(t)) \, d\lambda''_b(t)$ is $\nu$-integrable (Ch. V, §5, No. 3, Th. 1).

By the definition of $\lambda_b$, this function is $b \mapsto \int h(t) d\lambda_b(t)$, whence (loc. cit.) $\int h(t) d\mu(t) = \int d\nu(b) \int h(t) d\lambda_b(t)$, which proves that $\mu = \int \lambda_b d\nu(b)$.

To establish the second part of the statement, we remark that one can suppose that $r(b) > 0$ for all $b \in B$ (Ch. V, §5, No. 6, Prop. 10); set $\lambda'''_b = f \cdot \left( (r(b)/g(b)) \lambda'_b \right)$; one shows, as above, that for every function $h \in \mathscr{K}(T)$, the relation

implies

Therefore Th. 1 of No. 1, applied to $\mu''$ and $\nu''$, implies that for almost every $b \in B$, $\lambda'''_b = \lambda''_b$, whence $\lambda_b = r(b)\lambda'_b$.