In Folland's quantum field theory book he says:
Let $\{\sigma_t : t \in \mathbb{R}\}$ be a one-parameter group of measure-preserving diffeomorphisms of $\mathbb{R}^n$ whose orbits are (generically) unbounded, and suppose $F$ is a function on $\mathbb{R}^n$ that is invariant under these transformations. We wish to extract a finite and meaningful quantity from the divergent integral $\int F(x) d^nx$. One possibility is to find a hypersurface $M$ that is a cross-section for the orbits (perhaps after judiciously pruning away sets of measure zero) and consider the instead the integral $\int_M F(x) d\Sigma(x)$ where $d\Sigma$ is a surface measure on $M$.
I am confused on what is going on here. I am familiar with group actions and orbits, but what exactly does it mean to take a cross-section of an orbit? Furthermore, how do we define a surface measure on such an "orbit cross-section"?