Definition 2

Let T and B be two Polish locally compact spaces. Given a positive measure $\mu$ on T, a $\mu$-measurable mapping p of T into B, and a pseudo-image measure $\nu$ of $\mu$ under p, every $\nu$-adequate family $b \mapsto \lambda_b$ ($b \in B$) of positive measures on T having the properties b) and c) of Th. 2 is called a disintegration of $\mu$ relative to $\nu$.

When $p$ is $\mu$-proper and $\nu = p(\mu)$, the concept of disintegration relative to $p$ thus coincides with the concept of disintegration relative to $\nu$. Under the hypotheses of Th. 2, two disintegrations of $\mu$ relative to the same pseudo-image measure $\nu$ are equal almost everywhere for $\nu$.