Proof of 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

Each result cited below is universally quantified over the data in its own statement.

The natural number $d$ is read in $\mathbb{R}$ as in The Real Numbers: Standing Notation and Background §numbers wherever a real number is required. Write $\lambda=\lambda_{d}$ and $J(x)=\det DF(x)$; since $DF(x)$ is symmetric, it lies in $\mathcal{S}(d)$, and $0<J(x)$ 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. Metric notions on $\mathbb{R}^{d}$ (open sets, closures, boundedness, compactness) refer to the Euclidean distance $d_{E}$ as in Euclidean Space and Lebesgue Measure: Standing Notation §space, with $d_{E}(x,y)=\lVert x-y\rVert$; a real-valued function on a subset $A$ of $\mathbb{R}^{d}$ is called continuous when it is continuous on $A$ for $d_{E}$ and the absolute-value metric of $\mathbb{R}$. A function continuous on $\mathbb{R}^{d}$ is continuous on every subset $A$, directly from Continuous Map Between Metric Spaces. By Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps, a map $\mathbb{R}^{d}\to\mathbb{R}^{d}$ is Borel exactly when its components are, a continuous map $\mathbb{R}^{d}\to\mathbb{R}$ is Borel, and compositions of Borel maps are Borel; indicators of Borel sets, sums and products of real-valued Borel maps are Borel by claims 1 to 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. For $a\in\mathbb{R}^{d}$ and a positive real number $s$ put

the half-open cube with corner $a$ and side $s$, and let $Q(c,r)$ be the closed cube of A Continuously Differentiable Map Whose Jacobian Matrix is Close to the Identity Maps a Cube into a Slightly Larger Cube. Let $e=(1,\dots,1)\in\mathbb{R}^{d}$. Claims 1 to 5 use only the hypotheses of the theorem on $F$, so they hold for every map satisfying those hypotheses.

Claim 1 (Continuity and measurability; claim 1). (a) The components of $F$ and of $F^{-1}$ and the functions $\partial_{j}F_{i}$ and $\partial_{j}(F^{-1})_{i}$ are continuous on $\mathbb{R}^{d}$, and $F$ and $F^{-1}$ are continuous at every point in the Euclidean sense. (b) If $u:\mathbb{R}^{d}\to\mathbb{R}$ is continuous, then so are $u\circ F$ and $u\circ F^{-1}$. (c) $J$ is continuous. (d) $F$, $F^{-1}$ and $J$ are Borel, and $F(B)\in\mathcal{B}(\mathbb{R}^{d})$ for every $B\in\mathcal{B}(\mathbb{R}^{d})$.

(a) By clause 1 of C^k Maps on a Euclidean Open Set, read through Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, the components $F_{i}$, $(F^{-1})_{i}$ and the functions $\partial_{j}F_{i}$, $\partial_{j}(F^{-1})_{i}$ are continuous at every point of $\mathbb{R}^{d}$ in the Euclidean sense, hence continuous by claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions; and $F$ and $F^{-1}$ are continuous at every point in the Euclidean sense by Coordinatewise Criterion for Continuity of a Map Between Euclidean Spaces. (b) Let $x\in\mathbb{R}^{d}$. By claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions, $u$ is continuous at $F(x)$ in the Euclidean sense; by (a) and Composition of Continuous Euclidean Maps, $u\circ F$ is continuous at $x$ in the Euclidean sense, hence continuous at $x$ by the same claim 1. The same argument applies to $u\circ F^{-1}$. (c) By Determinant of a Real Square Matrix, $J(x)=\sum_{\sigma\in S_{d}}\operatorname{sgn}(\sigma)\prod_{i=1}^{d}\partial_{\sigma(i)}F_{i}(x)$ is a finite sum of constant multiples of finite products of the continuous functions of (a), hence continuous by claims 2, 3 and 4 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, applied by induction on the number of factors and of summands. (d) $F$, $F^{-1}$ and $J$ are Borel by (a), (c) and the preamble. Since $F$ is a bijection with inverse $F^{-1}$, $y\in F(B)$ holds exactly when $F^{-1}(y)\in B$, so $F(B)=(F^{-1})^{-1}(B)$ is Borel.

Claim 2 (Closed hulls of half-open cubes). For $a\in\mathbb{R}^{d}$ and positive $s$, the closure $\overline{P}(a,s)$ of $P(a,s)$ is compact, and $a\in P(a,s)\subseteq\overline{P}(a,s)$.

For $y\in P(a,s)$, $|a_{i}-y_{i}|\le s$ for every $i$, so $d_{E}(a,y)=\lVert a-y\rVert\le d\,s$ by Coordinate Bounds Control the Euclidean Norm. Hence $P(a,s)$ is bounded, and its closure is compact by The Closure of a Bounded Subset of $\mathbb{R}^n$ is Compact. A point of $P(a,s)$ lies in every open set containing it, so $P(a,s)\subseteq\overline{P}(a,s)$ by Closure of a Subset of a Topological Space; and $a\in P(a,s)$ as $0<s$.

Claim 3 (Images of half-open cubes). Let $a\in\mathbb{R}^{d}$, let $s$ be positive and $P=P(a,s)$. Then $F(P)\in\mathcal{B}(\mathbb{R}^{d})$, the number $I=\int_{\mathbb{R}^{d}}\mathbf{1}_{P}\,J\,d\lambda$ is finite, and $\lambda(F(P))\le I$.

Compactness data. $F(P)$ is Borel by Claim 1(d), $P$ being Borel by claim 1 of Uniform Grids on a Half-Open Box and Grid Hulls of a Compact Set in $\mathbb{R}^n$ (with $n=d$, corner $a$, $m=1$), which also gives $\lambda(P)=s^{d}$. Put $P^{+}=P(a-s\,e,3s)$ and let $K=\overline{P}(a-s\,e,3s)$, a nonempty compact set containing $P^{+}$ by Claim 2; note that $\{x:a_{i}\le x_{i}\le a_{i}+s\text{ for every }i\}\subseteq P^{+}$. For $i,j\in[d]$ let $e'_{ij}(x)=\partial_{j}(F^{-1})_{i}(F(x))$, the entry in row $i$ and column $j$ of $DF^{-1}(F(x))$. By Claim 1, the functions $\partial_{j}F_{i}$, $e'_{ij}$ and $J$ are continuous on $\mathbb{R}^{d}$, hence on $K$; by Heine-Cantor Theorem: Continuity on a Compact Subset Implies Uniform Continuity they are uniformly continuous on $K$, and by Extreme Value Theorem on a Compact Subset of a Metric Space each of the finitely many functions $e'_{ij}$ and $J$ attains a least and a greatest value on $K$. If $\ell\le v\le L'$ then $|v|\le|\ell|+|L'|$ by claims 3 and 6 of Properties of the Absolute Value in an Ordered Field; so with $\beta=1+\sum_{i,j}(|\ell_{ij}|+|L'_{ij}|)$, where $\ell_{ij}$ and $L'_{ij}$ are the least and greatest values of $e'_{ij}$ on $K$, we have $1\le\beta$ and $|e'_{ij}(x)|\le\beta$ for all $x\in K$ and $i,j$; and with $M_{J}$ the greatest value of $J$ on $K$, $\mathbf{1}_{P}J\le M_{J}\mathbf{1}_{P}$, so $I\le M_{J}\lambda(P)=M_{J}s^{d}<\infty$ by claim 1 of Linearity and Monotonicity of the Lebesgue Integral and The Integral of an Indicator Function is the Measure of the Set.

Order of choices. Let $\gamma$ be a positive real number. Put $\theta=\gamma/(2s^{d})$, and choose a real $\eta$ with $0<\eta\le1$ and $2^{d}\eta\,(I+\gamma/2)\le\gamma/2$. By the uniform continuity above, choose a positive real $\omega$ (the least of finitely many radii) such that all $x,x'\in K$ with $d_{E}(x,x')<\omega$ satisfy $|\partial_{j}F_{i}(x)-\partial_{j}F_{i}(x')|<\eta/(d^{2}\beta)$ for all $i,j$ and $|J(x)-J(x')|<\theta$. By claim 2 of The Archimedean Property of the Real Numbers, choose $N\in\mathbb{N}$ with $d\,s<N\omega$, and put $r=s/(2N)$, so that $d\,r<\omega$.

One cell. Let $Q_{\kappa}$, for $\kappa\in[N]^{d}$, be the cells $Q_{N,\kappa}$ of claim 1 of Uniform Grids on a Half-Open Box and Grid Hulls of a Compact Set in $\mathbb{R}^n$ (with $n=d$, corner $a$, side $s$, $m=N$): pairwise disjoint Borel sets with union $P$ and $\lambda(Q_{\kappa})=(s/N)^{d}$. Fix $\kappa$ and let $c\in\mathbb{R}^{d}$ have coordinates $c_{i}=a_{i}+(\kappa_{i}-\tfrac12)s/N$. Then $Q_{\kappa}\subseteq Q(c,r)\subseteq P^{+}\subseteq K$, and every $x\in Q(c,r)$ satisfies $d_{E}(x,c)\le d\,r<\omega$ by Coordinate Bounds Control the Euclidean Norm. Let $A=DF(c)$, symmetric and positive definite, and $A'=DF^{-1}(F(c))$, its inverse matrix, so $A'A=AA'=I_{d}$ and $|A'_{il}|=|e'_{il}(c)|\le\beta$. For $i\in[d]$ let $L_{i}(y)=\sum_{l=1}^{d}A'_{il}y_{l}$; by claim 1 of Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity (with $M=0_{d}$, $q=(A'_{i1},\dots,A'_{id})$, constant $0$), $L_{i}$ is of class $C^{2}$ on $\mathbb{R}^{d}$ with gradient $q$, i.e. $\partial_{l}L_{i}=A'_{il}$ (Gradient of a Real-Valued Function on a Euclidean Open Set), hence of class $C^{1}$ by claim 2 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. Let $G=(L_{1}\circ F,\dots,L_{d}\circ F)$, i.e. $G(x)=A'F(x)$. By claims 2 (with $k=1$) and 1 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$, the components of $G$ are of class $C^{1}$ and $\partial_{j}G_{i}(x)=\sum_{l}A'_{il}\,\partial_{j}F_{l}(x)$. As $\delta_{ij}=(A'A)_{ij}=\sum_{l}A'_{il}\,\partial_{j}F_{l}(c)$, claims 4 and 5 of Properties of the Absolute Value in an Ordered Field (the latter by induction over the finite sum) give, for $x\in Q(c,r)$ and all $i,j$,

By A Continuously Differentiable Map Whose Jacobian Matrix is Close to the Identity Maps a Cube into a Slightly Larger Cube §cube, $G(x)\in C:=Q(G(c),(1+\eta)r)$ for every $x\in Q(c,r)$. Since $AA'=I_{d}$, $F(x)=A\,G(x)$, so $F(Q_{\kappa})\subseteq A(C)=\{Az:z\in C\}$. The set $C$ is the product of the closed intervals $[G_{i}(c)-(1+\eta)r,\,G_{i}(c)+(1+\eta)r]$, which belong to $\mathcal{B}(\mathbb{R})$ and have Lebesgue measure $2(1+\eta)r$ by claim 4 of Existence of Lebesgue Measure on the Real Line; so $C$ is a Borel rectangle, it belongs to $\mathcal{B}(\mathbb{R}^{d})$ by claim 1 of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$ and claim 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, and $\lambda(C)=(1+\eta)^{d}(s/N)^{d}$ by Lebesgue Measure on $\mathbb{R}^n$. By Lebesgue Measure under a Triangular or Symmetric Positive Definite Linear Map §spd, $A(C)$ is Borel with $\lambda(A(C))=\det A\,\lambda(C)$. For $x\in Q_{\kappa}$, $\det A=J(c)<J(x)+\theta$. Hence, by claim 2 of Basic Properties of a Measure, The Integral of an Indicator Function is the Measure of the Set and claim 1 of Linearity and Monotonicity of the Lebesgue Integral,

Summation. As $F$ is injective, the Borel sets $F(Q_{\kappa})$ are pairwise disjoint with union $F(P)$, and $\sum_{\kappa}\mathbf{1}_{Q_{\kappa}}=\mathbf{1}_{P}$ pointwise. By claim 1 of Basic Properties of a Measure, claim 1 of Linearity and Monotonicity of the Lebesgue Integral (by induction over the finite set $[N]^{d}$) and The Integral of an Indicator Function is the Measure of the Set,

Since $0<\eta\le1$, induction on $d$ with $\eta^{2}\le\eta$ gives $(1+\eta)^{d}\le1+(2^{d}-1)\eta\le1+2^{d}\eta$, so by the choice of $\theta$ and $\eta$,

In particular $\lambda(F(P))$ is finite, and as $\gamma$ was arbitrary, $\lambda(F(P))\le I$ (were $\lambda(F(P))>I$, the choice $\gamma=\tfrac12(\lambda(F(P))-I)$ would contradict the display).

Claim 4 (Preimages of compact sets). Let $S\subseteq\mathbb{R}^{d}$ be nonempty and compact. Then there are $a\in\mathbb{R}^{d}$ and a positive $s$ with $F^{-1}(S)\subseteq P(a,s)$.

For each $i$, the component $(F^{-1})_{i}$ is continuous (Claim 1(a)), so its restriction to $S$ has the continuity property required in Extreme Value Theorem on a Compact Subset of a Metric Space, which gives real numbers $\ell_{i}\le u_{i}$ with $\ell_{i}\le(F^{-1})_{i}(y)\le u_{i}$ for every $y\in S$. With $t=\min_{i}\ell_{i}$, $s=\max_{i}u_{i}-t+1$ (positive, indeed $1\le s$, since $t=\ell_{i_{0}}\le u_{i_{0}}\le\max_{i}u_{i}$ for an index $i_{0}$ attaining the minimum) and $a=t\,e$ we get $a_{i}\le(F^{-1})_{i}(y)<a_{i}+s$ for all $y\in S$ and $i$.

Claim 5 (Lower bound for continuous integrands). Let $\psi:\mathbb{R}^{d}\to\mathbb{R}$ be continuous with $0\le\psi$, and let $S\subseteq\mathbb{R}^{d}$ be compact with $\psi(y)=0$ for every $y\notin S$. Then

Both integrands are nonnegative and Borel by Claim 1. If $S=\varnothing$, then $\psi=0$ and the left side is $0$ (claim 1 of Linearity and Monotonicity of the Lebesgue Integral with $c=0$). Otherwise take $P_{0}=P(a,s)\supseteq F^{-1}(S)$ from Claim 4, so that $S\subseteq F(P_{0})$, and let $K_{0}=\overline{P}(a,s)$, a nonempty compact set containing $P_{0}$ (Claim 2). The function $u=\psi\circ F$ is continuous (Claim 1(b)), hence uniformly continuous on $K_{0}$ by Heine-Cantor Theorem: Continuity on a Compact Subset Implies Uniform Continuity, and $J\le M_{0}$ on $K_{0}$ for some real $M_{0}$ by Extreme Value Theorem on a Compact Subset of a Metric Space. Let $\theta$ be a positive real number. Choose a positive $\omega$ such that $|u(x)-u(x')|<\theta$ for all $x,x'\in K_{0}$ with $d_{E}(x,x')<\omega$, and then, by claim 2 of The Archimedean Property of the Real Numbers, $N\in\mathbb{N}$ with $\sigma_{d}\,s<N\omega$, where $\sigma_{d}=\lVert e\rVert$. Let $Q_{\kappa}$, $\kappa\in[N]^{d}$, be the cells of claim 1 of Uniform Grids on a Half-Open Box and Grid Hulls of a Compact Set in $\mathbb{R}^n$ (with $n=d$, corner $a$, side $s$, $m=N$): pairwise disjoint Borel sets with union $P_{0}$, any two points of one cell being at distance at most $\sigma_{d}s/N<\omega$. Each $Q_{\kappa}$ is the half-open cube $P(a+\tfrac{s}{N}(\kappa-e),\tfrac{s}{N})$ and contains its corner $p_{\kappa}=a+\tfrac{s}{N}(\kappa-e)$. Put $m_{\kappa}=u(p_{\kappa})+\theta$, a nonnegative real number; then $u(x)<m_{\kappa}<u(x)+2\theta$ for $x\in Q_{\kappa}$. The sets $F(Q_{\kappa})$ are Borel (Claim 1(d)), pairwise disjoint with union $F(P_{0})\supseteq S$, and $\psi(F(x))=u(x)<m_{\kappa}$ for $x\in Q_{\kappa}$; as $\psi$ vanishes off $S$, $\psi\le\sum_{\kappa}m_{\kappa}\mathbf{1}_{F(Q_{\kappa})}$ pointwise. By claim 1 of Linearity and Monotonicity of the Lebesgue Integral (with induction over the finite set $[N]^{d}$), The Integral of an Indicator Function is the Measure of the Set, and Claim 3 applied to each half-open cube $Q_{\kappa}$,

where the third inequality uses $m_{\kappa}<u(x)+2\theta$ on $Q_{\kappa}$ and $0<J$, and the last uses $\mathbf{1}_{P_{0}}uJ\le uJ$, $\mathbf{1}_{P_{0}}J\le M_{0}\mathbf{1}_{P_{0}}$ and $\lambda(P_{0})=s^{d}$. If $\int uJ\,d\lambda=\infty$ there is nothing to prove; otherwise the display holds for every positive $\theta$, and were $\int\psi\,d\lambda>\int uJ\,d\lambda$, a $\theta$ with $2\theta M_{0}s^{d}<\int\psi\,d\lambda-\int uJ\,d\lambda$ (note $0<M_{0}$) would contradict it.

Claim 6 (The inverse map). $H=F^{-1}$ satisfies the hypotheses of the theorem, with inverse $F$, and $J(H(y))\det DH(y)=1$ for every $y\in\mathbb{R}^{d}$.

$H$ is a bijection with inverse $F$, and the components of $H$ and of $F$ are of class $C^{1}$. Let $y\in\mathbb{R}^{d}$ and $x=H(y)$, so $F(x)=y$. By hypothesis $DH(y)=DF^{-1}(F(x))$ is the inverse matrix of $DF(x)$. Since $DF(x)$ is symmetric and positive definite, Invertibility of Symmetric Positive Definite Matrices shows that it is invertible with symmetric positive definite inverse, which is $DH(y)$ by the uniqueness of the inverse in Inverse Matrix and Invertible Real Square Matrix. The identities $DF(x)\,DH(y)=I_{d}=DH(y)\,DF(x)$ also say that $DH^{-1}(H(y))=DF(x)$ is the inverse matrix of $DH(y)$. Finally $J(H(y))\det DH(y)=\det\bigl(DF(x)\,DH(y)\bigr)=\det I_{d}=1$ by The Determinant is Multiplicative and claim 1 of Row Properties of the Determinant.

Claim 7 (Continuous integrands). Under the hypotheses of Claim 5,

The inequality $\le$ is Claim 5. If $S=\varnothing$ both sides are $0$. Otherwise let $P_{0}\supseteq F^{-1}(S)$ and $K_{0}\supseteq P_{0}$ be as in the proof of Claim 5. The function $\psi'=(\psi\circ F)J$ is continuous (Claim 1(b), (c) and claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space) and nonnegative, and vanishes outside the compact set $K_{0}$: if $x\notin K_{0}$ then $x\notin F^{-1}(S)$, so $F(x)\notin S$ and $\psi(F(x))=0$. By Claim 6, Claim 5 applies to the map $H=F^{-1}$ and the function $\psi'$ with the compact set $K_{0}$, giving $\int\psi'\,d\lambda\le\int(\psi'\circ H)\det DH\,d\lambda$. Since $F(H(y))=y$ and $J(H(y))\det DH(y)=1$ (Claim 6), $(\psi'\circ H)(y)\det DH(y)=\psi(y)$ for every $y$, which gives $\ge$.

Claim 8 (Lebesgue measure through $F$). For every $A\in\mathcal{B}(\mathbb{R}^{d})$, $\lambda(A)=\int_{\mathbb{R}^{d}}(\mathbf{1}_{A}\circ F)\,J\,d\lambda$.

A weight. Let $h=g_{1}$ be the Gaussian of The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails in dimension $q=d$, which is the Gaussian smoothing weight $\varphi_{1}$ of claim 1 of The Gaussian Smoothing Weight: Normalization, Derivatives, Exponential Tilting, Moments, and First-Order Remainder with $m=d$ and $\eta=1$; by that claim (with $a=0_{\mathbb{R}^{d}}$), $0<h(z)$ for every $z$ and $\int h\,d\lambda=1$. By The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails §derivatives, $h$ is smooth, hence continuous by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous; $1/h$ is continuous by claim 2 of Continuity of the Reciprocal of a Nonvanishing Real-Valued Function on a Metric Space; both are Borel. The function $k=(h\circ F)J$ is continuous (Claim 1(b), (c) and claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space), positive and Borel. Let $\nu_{2}$ and $\mu_{k}$ be the measures with densities $h$ and $k$ with respect to $\lambda$ (claim 3 of Image Measures, Measures with Densities, and Change of Variables), and $\nu_{1}$ the image measure of $\mu_{k}$ under the Borel map $F$ (claim 1 there). By claims 2 and 3 there, for every Borel $g:\mathbb{R}^{d}\to[0,\infty]$,

Both are measures on $\mathcal{B}(\mathbb{R}^{d})$, the Borel $\sigma$-algebra of $(\mathbb{R}^{d},d_{E})$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces), and $\nu_{2}(\mathbb{R}^{d})=\int h\,d\lambda=1$ by ($*$) and The Integral of an Indicator Function is the Measure of the Set.

(i) Nonnegative continuous integrands. If $\psi:\mathbb{R}^{d}\to\mathbb{R}$ is continuous, $0\le\psi$, and $\psi$ vanishes outside a compact set $S$, then $\psi h$ has the same properties (claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space), and by ($*$) and Claim 7 applied to $\psi h$,

(ii) $\nu_{1}$ is finite. For $m\in\mathbb{N}$ let $A_{m}=\{z:m<\lVert z\rVert\}$, which is Borel by The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails §moments (with $s=1$, $R=m$), and $E_{m}=\mathbb{R}^{d}\setminus A_{m}=\{z:\lVert z\rVert\le m\}$, Borel by Sigma-Algebra and Measurable Space. Let $\chi_{m}$ be the cutoff of Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff with $q=d$ and $R=m$: it is smooth, hence continuous (claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous), compactly supported, so it vanishes outside its compact support by claim 1 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set; moreover $0\le\chi_{m}\le1$ and $\chi_{m}=1$ on $E_{m}$, so $\mathbf{1}_{E_{m}}\le\chi_{m}\le1$. By The Integral of an Indicator Function is the Measure of the Set, claim 1 of Linearity and Monotonicity of the Lebesgue Integral and (i),

The sets $E_{m}$ increase with $m$, and their union is $\mathbb{R}^{d}$ because every $z$ has $\lVert z\rVert<m$ for some $m$ by claim 1 of The Archimedean Property of the Real Numbers. By claim 5 of Basic Properties of a Measure, $\nu_{1}(\mathbb{R}^{d})$ is the least upper bound of the numbers $\nu_{1}(E_{m})$, so $\nu_{1}(\mathbb{R}^{d})\le1$.

(iii) $\nu_{1}=\nu_{2}$. Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ (Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §test, Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §space). It is smooth, hence continuous, and compactly supported; by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set there is a positive $R$ with $\psi(x)=0$ whenever $R<\lVert x\rVert$, and by claim 1 of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable and Bounded Real-Valued Function on a Set there is a real $M\ge0$ with $|\psi(x)|\le M$ for all $x$. Let $\chi=\chi_{R}$ be the cutoff of Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff with $q=d$, and put $\psi_{1}=\psi+M\chi$ and $\psi_{2}=M\chi$, continuous by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space. Both are nonnegative: if $\lVert x\rVert\le R$ then $\chi(x)=1$ and $0\le\psi(x)+M$ by claim 6 of Properties of the Absolute Value in an Ordered Field, and otherwise $\psi(x)=0$. Both vanish outside the compact support of $\chi$: there $\chi(x)=0$ by claim 1 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set, so $R<\lVert x\rVert$ (as $\chi=1$ on $\lVert x\rVert\le R$) and $\psi(x)=0$. By (i), $\int\psi_{i}\,d\nu_{1}=\int\psi_{i}\,d\nu_{2}$ for $i=1,2$, and these integrals are at most $2M$ by (ii), $\psi_{i}\le2M$, claim 1 of Linearity and Monotonicity of the Lebesgue Integral and The Integral of an Indicator Function is the Measure of the Set. So $\psi_{1}$, $\psi_{2}$ are integrable with respect to $\nu_{1}$ and $\nu_{2}$, their integrals in the sense of Integrable Function and the Lebesgue Integral being the ones above since their negative parts are $0$; by claim 2 of Linearity and Monotonicity of the Lebesgue Integral applied to $\psi=\psi_{1}-\psi_{2}$,

Since $\nu_{1}$ and $\nu_{2}$ are finite Borel measures on $(\mathbb{R}^{d},d_{E})$, Test Functions Approximate Bounded Continuous Functions Pointwise, Determine a Finite Borel Measure, and Detect a Vanishing Density §determination gives $\nu_{1}(B)=\nu_{2}(B)$ for every $B\in\mathcal{B}(\mathbb{R}^{d})$.

(iv) Conclusion. Let $A\in\mathcal{B}(\mathbb{R}^{d})$ and $u_{A}=\mathbf{1}_{A}\cdot(1/h)$, Borel and nonnegative. Pointwise $u_{A}h=\mathbf{1}_{A}$ and $(u_{A}\circ F)\,k=(\mathbf{1}_{A}\circ F)\,J$. By The Integral of an Indicator Function is the Measure of the Set, ($*$) and (iii),

Step 1 (Proof of claim 2). Let $\rho_{0}$ be the measure with density $J$ (Borel and positive) with respect to $\lambda$, and $\rho_{F}$ its image measure under $F$ (claims 3 and 1 of Image Measures, Measures with Densities, and Change of Variables). Since $\mathbf{1}_{F^{-1}(A)}=\mathbf{1}_{A}\circ F$, Claim 8 gives $\rho_{F}(A)=\rho_{0}(F^{-1}(A))=\int(\mathbf{1}_{A}\circ F)J\,d\lambda=\lambda(A)$ for every Borel $A$, so $\rho_{F}=\lambda$. For Borel $f:\mathbb{R}^{d}\to[0,\infty]$, claims 2 and 3 of Image Measures, Measures with Densities, and Change of Variables give

For Borel $f:\mathbb{R}^{d}\to\mathbb{R}$, the same claims say that $f$ is integrable with respect to $\rho_{F}=\lambda$ if and only if $f\circ F$ is integrable with respect to $\rho_{0}$, if and only if $(f\circ F)J$ is integrable with respect to $\lambda$, and that then the same chain of equalities holds in $\mathbb{R}$.

Step 2 (Proof of claim 3). Let $\rho$ be a density of $\mu$ with respect to $\lambda$ in the sense of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities: $\rho$ is Borel, $0\le\rho$, and $\mu(A)=\int\mathbf{1}_{A}\rho\,d\lambda$ for every Borel $A$. Since $F$ is Borel, $F_{\#}\mu\in\mathcal{P}(\mathbb{R}^{d})$ is defined, with $F_{\#}\mu(B)=\mu(F^{-1}(B))$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward); it is a finite measure. The function $J\circ F^{-1}$ is continuous (Claim 1(b), (c)) and positive, so $1/(J\circ F^{-1})$ is continuous by claim 2 of Continuity of the Reciprocal of a Nonvanishing Real-Valued Function on a Metric Space, hence Borel; $\rho\circ F^{-1}$ is Borel; so $\tilde\rho=(\rho\circ F^{-1})\cdot\bigl(1/(J\circ F^{-1})\bigr)$ is Borel, and it is nonnegative. Let $B\in\mathcal{B}(\mathbb{R}^{d})$. For each $x$, $F^{-1}(F(x))=x$ gives $\tilde\rho(F(x))\,J(x)=\rho(x)$, so pointwise $\bigl((\mathbf{1}_{B}\tilde\rho)\circ F\bigr)J=\mathbf{1}_{F^{-1}(B)}\,\rho$. By Step 1 applied to the Borel function $\mathbf{1}_{B}\tilde\rho\ge0$,

Hence $\tilde\rho$ is a density of $F_{\#}\mu$ with respect to $\lambda$.