Under certain conditions on the functions $g:\mathbb{R}^n \to \mathbb{R}$ and $f:\mathbb{R} \to \mathbb{R}$ involved I have seen formulas such as $ \int \limits _{\mathbb{R}^n} f\circ g (x) \,dx = \int f(t) \, d \nu (t) , $ where $ \nu (t) = - \int \limits _{g(x) \ge t} \,dx . $ Sometimes this formula comes without the minus sign and the inequality reversed, which I suppose is so that it is an increasing function. Is the integration in $t$ for all $t$ or just positive $t$?
It appears clear that this is related to the general change of variables formula $ \int f\circ g \, d\mu = \int f \, d\nu $ where $\nu (B) = \mu (g^{-1}(B))$, but I must admit I can't see the exact chain of equalities linking them together.
Is there any good reference explaining these things, or is it really easy to see?