Pullback Invariance of the Integral under Orientation-Preserving Smooth Diffeomorphisms

Let $n\in$ $\mathbb{N}$, let $\Omega,\Omega'$ be admissible domains in Euclidean space $\mathbb{R}^n$ in the sense of Continuous n-Form, Support, and Zero Extension on a Euclidean or Half-Space Domain, and let $F:\Omega\to\Omega'$ be an orientation-preserving smooth diffeomorphism. Let $\omega$ be a continuous differential $n$-form on $\Omega'$ that is compactly supported in $\Omega'$, in the sense of Continuous n-Form, Support, and Zero Extension on a Euclidean or Half-Space Domain.

Then the pullback $F^{*}\omega$ is a continuous differential $n$-form on $\Omega$ that is compactly supported in $\Omega$, and

$$\int_{\Omega}F^{*}\omega=\int_{\Omega'}\omega,$$

where both sides are integrals in the sense of Integral of a Compactly Supported Continuous n-Form on a Euclidean or Half-Space Domain.