Change of variables formula in terms of Lebesgue measure

Statement

The following theorem allows us to relate integrals with respect to Lebesgue measure to an equivalent integral with respect to the pullback measure under a parameterization G. The proof is due to approximations of the Jordan content.

Suppose that $\Omega$ is an open subset of $\mathbb{R}^n$ and $G:\Omega \to \mathbb{R}^n$ is a $C^1$ diffeomorphism.

If $f$ is a Lebesgue measurable function on $G(\Omega)$, then $f \circ G$ is Lebesgue measurable on $\Omega$. If $f \geq 0$ or $f\in L^1(G(\Omega),m),$ then $\int_{G(\Omega)} f(x) dx = \int_\Omega f\circ G(x)|\text{det}D_xG|dx$.

If $E\subset \Omega$ and $E$ is Lebesgue measurable, then $G(E)$ is Lebesgue measurable, then $m(G(E)) = \int_E |\text{det}D_xG| dx$.

As a corollary of this theorem, we may compute the Radon–Nikodym derivatives of both the pullback and pushforward measures of $m$ under $T$.