I am a physicist, and I have the following problem. Consider a locally compact group G acting over a measure space $(X, {\cal B}, \mu)$, and assume that $\mu$ is G-invariant. My problem is how to "quotient" the measure $\mu$ for obtaining a measure $\mu/G$ on the quotient space $X/G$, i.e., the space wose elements are the orbits of G. The simple answer consisiting of defining $\mu/G(\Delta):=\mu(\Delta)$ for $\Delta \in X/G$ is not appropriate, as one can see from the following example.

Let $X:=\mathbb{R}^2$ be the configuration space of two one-dimensional particles, let $\mu$ be the Lebesgue measure, and let $G:=\mathbb{R}$ be the group of translations: $a(x, y)=(x + a, y + a)$ for $a \in G$ and $(x, y) \in X$. The orbit of a point $(x, y)$ is the line at $45^°$ passing for $(x, y)$. It is easy to see that the Lebesgue measure of any set of orbits is either $0$ or $\infty$ .

My tentative answer is the following. Let S be a section of the partition of the orbits, i.e., a subset of X composed by an element for every orbit. Let the map $h_S:G \times X/G \to X$ be defined as follows: $h_S(g, \xi)=g s_\xi$, where $s_\xi$ is the element of the section $S$ belonging to the orbit $\xi$. It is easy to see that $h_S$ is a bijection between $G \times X/G$ and $X$. It is also easy to see that a measure $\nu_S$ on $X/G$ exists such that $\mu=h_S(\alpha \times \nu_S)$, where $\alpha$ is the Haar measure on G. The measure $\nu_S$ is the measure I am looking for. The problem is to prove rigorously that it does not depend on the chosen section S.

I guess that this problem has already been addressed. Can anybody give me some reference?