If $\mu$ is a measure on $X$ and, for each $x$, $\nu_x$ is a measure on $Y$, what is the measure $\int \nu_x d\mu(x)$? > If $\mu$ is a measure on $X$ and, for each $x$, $\nu_x$ is a measure on $Y$, what is the measure $\int \nu_x d\mu(x)$? We have two measure spaces $X$ and $Y$, and we consider their product $Z = X \times Y$. The space $X$ comes with a measure $\mu$, and $Y$ comes with a collection of measures $\nu_x$ for each $x \in X$. Given any measurable $f : X \times Y \to \mathbb{R}$, we consider the natural integral $$\int_X \int_Y f(x, y) d\nu_x(y) d\mu(x).$$ If everything is nice, we should get a measure on $Z$. I have two questions: 1. What is this measure called? 2. Is there a reverse process where we start with measures on $Z$ and $X$ and factor the $Z$-measure over the $X$-measure and get a measure on each fibre $\\{x\\} \times Y$?

2. The answer is "Yes, under suitable conditions". This reverse process is called the disintegration of the measure on $Z$. See < for example. The notion is very closely related to conditional distributions. For more on this see the article of Chang and Pollard "Conditioning as Disintegration" [ _Statist. Neerlandica_ , vol. 51 (1997) pp. 287–317] < .