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  "body": "\n## Statement\n\nThe 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.\n\n\u003e Suppose that $\\Omega$ is an open subset of $\\mathbb{R}^n$ and $G:\\Omega \\to \\mathbb{R}^n$ is a $C^1$ diffeomorphism.\n\u003e\n\u003e 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 $\\int_{G(\\Omega)} f(x) dx = \\int_\\Omega f\\circ G(x)|\\text{det}D_xG|dx$.\n\u003e\n\u003e If $E\\subset \\Omega$ and $E$ is Lebesgue measurable, then $G(E)$ is Lebesgue measurable, then $m(G(E)) = \\int_E |\\text{det}D_xG| dx$.\n\nAs a corollary of this theorem, we may compute the Radon–Nikodym derivatives of both the pullback and pushforward measures of $m$ under $T$.\n",
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