Change of Variables for the Lebesgue Integral under a Continuously Differentiable Bijection of Euclidean Space with Symmetric Positive Definite Jacobian Matrix, and the Density of a Push-Forward

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, let $d$ be a natural number with $1\le d$, let $\lambda_{d}$ be Lebesgue measure on $\mathcal{B}(\mathbb{R}^{d})$, and let densities with respect to $\lambda_{d}$ be those of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities. The set $\mathbb{R}^{d}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. For a map $G:\mathbb{R}^{d}\to\mathbb{R}^{d}$ whose components $G_{1},\dots,G_{d}$ are of class $C^{1}$ on $\mathbb{R}^{d}$ and for $x\in\mathbb{R}^{d}$, $DG(x)$ denotes its Jacobian matrix at $x$, whose entry in row $i$ and column $j$ is $\partial_{j}G_{i}(x)$. Positive definiteness and the inverse matrix are those of those definitions, and $\det$ is the determinant.

Let $F:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be a bijection with inverse $F^{-1}$ such that the components of $F$ and of $F^{-1}$ are of class $C^{1}$ on $\mathbb{R}^{d}$ and, for every $x\in\mathbb{R}^{d}$, the matrix $DF(x)$ is symmetric and positive definite and $D(F^{-1})(F(x))$, the matrix of partial derivatives of the components of $F^{-1}$ at $F(x)$, is the inverse matrix of $DF(x)$. Then $0<\det DF(x)$ for every $x$ by Determinants of Positive Definite Matrices: Positivity, the Bound $\log\det A\le\mathrm{tr}\,A-d$, Bounds under Pinching, and the Expansion of $\det(I+tB)$ §positive.

1. (Regularity)¶ $F$ and $F^{-1}$ are continuous and Borel, and the function $x\mapsto\det DF(x)$ is continuous on $\mathbb{R}^{d}$, hence Borel.

2. (Change of variables)¶ For every Borel $f:\mathbb{R}^{d}\to[0,\infty]$,

A Borel $f:\mathbb{R}^{d}\to\mathbb{R}$ is integrable with respect to $\lambda_{d}$ if and only if $(f\circ F)\det DF$ is, and then the displayed identity holds in $\mathbb{R}$.

3. (Density of a push-forward)¶ Let $\mu\in\mathcal{P}(\mathbb{R}^{d})$ have a density $\rho$ with respect to $\lambda_{d}$. Then the function

is a density of $F_{\#}\mu$ with respect to $\lambda_{d}$.