Theorem 1

*Let T and B be two locally compact spaces having countable bases, $\mu$ a positive measure on T, p a $\mu$-proper mapping (Ch. V, §6, No. 1, Def. 1) of T into B, and $\nu = p(\mu)$ the image of $\mu$ under p. Then there exists a $\nu$-adequate family ($§ 1$, No. 1, Example) $b \mapsto \lambda_b$ ($b \in B$) of positive measures on T, having the following properties:

a) $\| \lambda_b \| = 1$ for all $b \in p(T)$;

b) $\lambda_b$ is concentrated on the set $\overline{p}^{-1}(b)$ (Ch. V, §5, No. 7, Def. 4) for all $b \in B$; in particular, $\lambda_b = 0$ for $b \notin p(T)$;

c) $\mu = \int \lambda_b d\nu(b)$.

Moreover, if $b \mapsto \lambda'_b$ ($b \in B$) is a second $\nu$-adequate family of positive measures on T having the properties b) and c), then $\lambda'_b = \lambda_b$ almost everywhere in B for the measure $\nu$.

1) Uniqueness. For every function $f \in \mathscr{K}(B)$, $f \circ p$ is $\mu$-integrable since p is $\mu$-proper (Ch. V, §6, No. 2, Th. 1); for every function $g \in \mathscr{K}(T)$, the function $t \mapsto g(t)f(p(t))$ is therefore $\mu$-integrable. It follows (Ch. V, §3, No. 3, Th. 1) that for almost every $b \in B$, the function $t \mapsto g(t)f(p(t))$ is $\lambda_b$-integrable and that

But since $\lambda_b$ is concentrated on $\overline{p}^{-1}(b)$, we have, for every $b \in B$, $f(p(t)) = f(b)$ almost everywhere for $\lambda_b$, therefore the second member of (1) is equal to $\int f(b)\langle g, \lambda_b \rangle d\nu(b)$. The analogous formula for $\lambda'_b$ also holds; consequently $\int f(b)\langle g, \lambda_b \rangle d\nu(b) = \int f(b)\langle g, \lambda'_b \rangle d\nu(b)$ for all $f \in \mathscr{K}(B)$ and $g \in \mathscr{K}(T)$. In other words, the two mappings $b \mapsto \lambda_b$ and $b \mapsto \lambda'_b$ of B into $\mathscr{M}(T)$ are equal scalarly locally almost everywhere for $\nu$, hence equal almost everywhere for $\nu$ (Lemma 1 and §1, No. 1, Remark 2).

2) Provisional definition of the family $b \mapsto \lambda_b$. For every function $f \in \mathscr{L}^1(\nu)$, $f \circ p$ is $\mu$-integrable (Ch. V, §6, No. 2, Th. 1), therefore $(f \circ p) \cdot \mu$ is a bounded measure on T, and

(Ch. IV, §4, No. 7, Prop. 12; Ch. V, §5, No. 3, Th. 1 and §6, No. 2, Th. 1). It follows that $(f \circ p) \cdot \mu$ depends only on the class $\tilde{f}$ of $f$ in $L^1(\nu)$ and that $\tilde{f} \mapsto (f \circ p) \cdot \mu$ is an isometric linear mapping of $L^1(\nu)$ into the Banach space $\mathscr{M}^1(T)$ of bounded measures on T, the strong dual of the Banach space $\mathscr{K}(T)$, which is separable (Lemma 2). By the Dunford–Pettis theorem (§2, No. 5, Cor. 2 of Th. 1) there exists a mapping $b \mapsto \lambda_b$ of B into the unit ball of $\mathscr{M}^1(T)$, scalarly $\nu$-measurable (for the topology $\sigma(\mathscr{M}^1(T), \overline{\mathscr{K}(T)})$) and such that, for every function $f \in \mathscr{L}^1(\nu)$,

which may also be written, for every function $g \in \overline{\mathscr{K}(T)}$

If $f \geq 0$ and $g \geq 0$, the first member of (3) is $\geq 0$, which proves that for every function $g \geq 0$ in $\mathscr{K}(T)$, the measure $(\int g(t) \, d\lambda_b(t)) \cdot \nu$ is $\geq 0$, hence that $\int g(t) \, d\lambda_b(t) \geq 0$ except for $b$ belonging to a $\nu$-negligible set $N(g)$ (Ch. V, §5, No. 3, Cor. 3 of Prop. 3). Now, there exists a dense sequence $(g_n)$ in the space $\mathscr{K}_+(T)$ of functions $\geq 0$ of $\mathscr{K}(T)$ (Lemma 1). The union N of the $N(g_n)$ is $\nu$-negligible and, for $b \notin N$, we have $\int g_n(t) \, d\lambda_b(t) \geq 0$ for all $n$, therefore $\int g(t) \, d\lambda_b(t) \geq 0$ for every function $g \in \mathscr{K}_+(T)$, in other words $\lambda_b \geq 0$.

This being so, we may replace $\lambda_b$ by 0 for every $b \in N$ without altering the validity of (3); we can therefore assume this modification to have been made, so that $\lambda_b \geq 0$ for every $b \in B$.

3) Extensions of the formula (3).

$\alpha)$ For every function $f \in \mathscr{L}^1(\nu)$, it follows from (3) that the mapping $b \mapsto \lambda_b$ of B into $\mathscr{M}(T)$ is scalarly integrable for the measure $|f \cdot \nu|$ and the topology $\sigma(\mathscr{M}(T), \mathscr{K}(T))$, therefore (Lemma 3) the family $b \mapsto \lambda_b$ is $|f \cdot \nu|$-*adequate*. Now let $g$ be a numerical function defined on T, integrable for the measure $|(f \circ p) \cdot \mu|$, that is (Ch. V, §5, No. 3, Th. 1), such that $t \mapsto g(t)f((p(t)))$ is $\mu$-integrable; it then follows from (2), from Th. 1 of Ch. V, §3, No. 3 and from Th. 1 of Ch. V, §5, No. 3 that, for almost every $b \in B$, $g$ is integrable for $\lambda_b$, that the function (defined almost everywhere) $b \mapsto \int g(t)\, d\lambda_b(t)$ is integrable for $|f \cdot \nu|$, and that the formula (3) is again valid.

$\beta)$ For every function $g \in \overline{\mathscr{K}(T)}$, it follows from (3), applied to $f \in \mathscr{K}(B)$, that the mapping $p$ is proper for the measure $|g \cdot \mu|$ (Ch. V, §6, No. 1, Def. 1) and the image under $p$ of the measure $g \cdot \mu$ is the measure with density $b \mapsto \int g(t)\, d\lambda_b(t)$ with respect to $\nu$. If $f$ is then taken to be a function such that $f \circ p$ is integrable for the measure $|g \cdot \mu|$, that is, such that $t \mapsto g(t)f(p(t))$ is $\mu$-integrable (Ch. V, §5, No. 3, Th. 1), the formula (3) is again valid (Ch. V, §6, No. 2, Th. 1).

4) Properties of the family $b \mapsto \lambda_b$. By the property $\beta)$, we can apply formula (3) by taking $f = 1,\ g \in \mathscr{K}(T)$; this proves that $b \mapsto \lambda_b$ is scalarly $\nu$-integrable (for the topology $\sigma(\mathscr{M}(T), \mathscr{K}(T))$), hence is $\nu$-*adequate* (Lemma 3), and that $\mu = \int \lambda_b\, d\nu(b)$.

Now let $\psi$ be any function in $\mathscr{K}(B)$; the conditions of property $\alpha)$ are fulfilled by taking $f \in \mathscr{K}(B)$ and $g = \psi \circ p$, because the function $\psi(p(t))f(p(t))$ is $\mu$-integrable since $f\psi$ belongs to $\mathscr{K}(B)$ and $p$ is $\mu$-proper. Then $\psi \circ p$ is $\lambda_b$-integrable for almost every $b \in B$, and

but the first member is by definition $\int f(b)\psi(b)\, d\nu(b)$. We therefore see that for every function $\psi \in \mathscr{K}(B)$, the measure $\psi \cdot \nu$ and the measure with density $b \mapsto \int \psi(p(t))\, d\lambda_b(t)$ are identical. Consequently (Ch. V, §5, No. 3, Cor. 2 of Prop. 3) there exists a $\nu$-negligible set $N'(\psi)$ such that, for every $b \notin N'(\psi)$, the function $\psi \circ p$ is $\lambda_b$-integrable and $\psi(b) = \int \psi(p(t))\, d\lambda_b(t)$.

Let S be a countable subset of $\mathscr{K}(B)$ having the properties stated in Lemma 1 (with $Y = B$), and let $N'$ be the $\nu$-negligible set that is the union of the $N'(\psi)$ for $\psi \in S$. Every function $\psi \geq 0$ of $\mathscr{K}(B)$ is the uniform limit of a sequence $(\psi_n)$ of elements of S with $\psi_n \leq \psi_0$. Consequently for $b \notin N'$, Lebesgue's theorem shows on the one hand that $\psi \circ p$ is $\lambda_b$-integrable, in other words that $p$ is $\lambda_b$-proper, and on the other hand that $\psi(b) = \int \psi(p(t))\, d\lambda_b(t)$. In other terms, the mappings $b \mapsto \varepsilon_b$ and $b \mapsto p(\lambda_b)$ of B into $\mathscr{M}(B)$ (the latter being defined almost everywhere) are scalarly almost everywhere equal for $\nu$ (and for the topology $\sigma(\mathscr{M}(B), \mathscr{K}(B))$); it follows that these mappings are equal almost everywhere for $\nu$ (Lemma 1 and §1, No. 1, Remark 2). Finally, if $p(\lambda_b) = \varepsilon_b$, the set $B - \{b\}$ is $\varepsilon_b$-negligible, therefore the set $T - \bar{p}^{-1}(B)$ is $\lambda_b$-negligible (Ch. V, §6, No. 2, Cor. 2 of Prop. 2), in other words $\lambda_b$ is concentrated on $\overline{p}^{-1}(b)$; and, on the other hand, $\| \lambda_b \| = \int d\lambda_b = \int d((p(\lambda_b))) = \| \varepsilon_b \| = 1$ (Ch. V, §6, No. 2, Th. 1).

5) Modifications of the family $b \mapsto \lambda_b$. We have thus defined a $\nu$-adequate family $b \mapsto \lambda_b$ of measures $\geqslant 0$ on T, satisfying condition c) of the statement and such that, for almost every $b \in B$, $p$ is $\lambda_b$-proper, and $\lambda_b$ is concentrated on $\overline{p}^{-1}(b)$ and has norm 1. Let $N''$ be the $\nu$-negligible set of points $b \in B$ where one of the last three properties is not verified; we can then modify $\lambda_b$ in the following manner. If $b \in B - p(T)$, take $\lambda_b = 0$; if $b \in p(T) \cap N''$, take $\lambda_b = \varepsilon_{\xi(b)}$, where $\xi(b)$ is any point of $\overline{p}^{-1}(b)$. Since $B - p(T)$ is $\nu$-negligible (Ch. V, §6, No. 2, Cor. 3 of Prop. 2), we have only modified $\lambda_b$ at the points of a negligible set, consequently the family $b \mapsto \lambda_b$ is still $\nu$-adequate and has the property c); moreover, it now satisfies a) and b), which completes the proof.

Every $\nu$-adequate family $b \mapsto \lambda_b$ of positive measures on T, having the properties b) and c) of Th. 1, is said to be a disintegration of the measure $\mu$, relative to the $\mu$-proper mapping $p$.