Equality of expectation value integral over coordinate space and over energy

Dear all,

I'm wondering, how one could justify mathematically the equality
[itex]\int O(E(\vec{x}_1,...\vec{x}_N)) exp(-\beta E(\vec{x}_1,...,\vec{x}_N)) d\vec{x}_1...d\vec{x}_N[/itex] = [itex]\int g(E) O(E) exp(-\beta E) dE[/itex]

where O(E(x)) is an observable and g(E) the density of states.

Is there a mathematical justification for the equality?

best,
derivator