In the context of statistical mechanics (so functions of real output are often defined on a 2n dimensional vector space called the phase space, of whose measure I will indicate with $d\Gamma$ ), the following was written $$\int_{H<E} d\Gamma= \int_0^E \int \delta(H(p,q)-\epsilon) d\Gamma d\epsilon$$ where H is a nowhere negative function of real output defined on all of phase space and the domain of integration of the first integral is to be understood as all points of phase space where H is less than E and when not specified the integration occurs over all phase space.
As far as I understood it, this is equivalent to the statement that $$\int_0^E \int_{H=\epsilon} d\Gamma d\epsilon$$ is the first lefthand side integral I wrote. Can anyone help me make sense of this? I of course understand the intuition but how exactly does one "integrate over the domain of integration" in this manner? I'm interested in the rigorous way of doing this.