Proof of The Cameron-Martin Translation Formula for a Diagonal Gaussian Measure on a Hilbert Space

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

Throughout, $h\in H_{c}$ is fixed, coordinates are $x_{k}=\langle x,e_{k}\rangle$ as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, and for $v\in X$ and $m\in\mathbb{N}$ we write

$$s_{v,m}=\sum_{k=1}^{m}\frac{v_{k}^{2}}{c_{k}},\qquad W_{v,m}:X\to\mathbb{R},\quad W_{v,m}(x)=\exp\Bigl(\ell_{v,m}(x)-\tfrac12 s_{v,m}\Bigr),$$

where $\ell_{v,m}$ is the $m$-th Paley-Wiener partial sum of $v$ relative to $c$, so $\ell_{v,m}(x)=\sum_{k=1}^{m}v_{k}x_{k}/c_{k}$. We abbreviate $s_{m}=s_{h,m}$ and $W_{m}=W_{h,m}$. For $m\in\mathbb{N}$ the truncation $c^{(m)}$ is a variance vector in $\mathbb{R}^{m}$ by Variance Sequences and Their Truncations §truncations, with scaling map $S_{c^{(m)}}$, weighted square $|a|_{c^{(m)}}^{2}$ ($a\in\mathbb{R}^{m}$) and density $\rho_{c^{(m)}}$ as in The Diagonal Gaussian Density on Euclidean Space and Its Notation §scaling and The Diagonal Gaussian Density on Euclidean Space and Its Notation §density.

Step 0 (Preliminaries). (a) The $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{m})$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures, the Borel $\sigma$-algebra of $(\mathbb{R}^{m},d_{E})$, coincides by The Borel $\sigma$-Algebras of Euclidean Space and of the Euclidean Metric Coincide with the Borel $\sigma$-algebra of Euclidean space on which Lebesgue measure $\lambda_{m}$, the measure $\gamma_{c^{(m)}}$ of Diagonal Gaussian Measures on Euclidean Space §measure and the statements of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$ are formulated; and the Borel $\sigma$-algebra of $(\mathbb{R},d_{\mathbb{R}})$ is the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ of the real line by claim 2 of Borel Measurability and Bounded Integration on a Metric Space. So "Borel" and "measurable" mean the same thing for real-valued functions below, and by Measure Spaces and the Lebesgue Integral: Standing Notation §measurable a measurable function with values in $[0,\infty)$ is also measurable as a $[0,\infty]$-valued function.

(b) By claim 2 of Derivative and Continuity of the Scaled Exponential Function, taken with the constant there equal to $1$ (so that the function there is $\exp$), $\exp$ is continuous from $(\mathbb{R},d_{\mathbb{R}})$ to itself; hence $\exp(t_{k})\to\exp(t)$ whenever $t_{k}\to t$ in $\mathbb{R}$, by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential, since convergence in $\mathbb{R}$ is convergence in $(\mathbb{R},d_{\mathbb{R}})$; and $\exp$ is Borel by claims 3 and 2 of Borel Measurability and Bounded Integration on a Metric Space. Consequently, for every measurable space $(\Omega,\mathcal{F})$ and every measurable $u:\Omega\to\mathbb{R}$, the function $\exp\circ u$ is measurable by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, and it is positive by claim 2 of Basic Properties of the Exponential Function.

(c) For $m\in\mathbb{N}$, $y\in\mathbb{R}^{m}$ and $j\in\mathbb{N}$, linearity of the inner product and orthonormality of $(e_{k})$ (Orthonormal Basis of a Real Hilbert Space §basis, Orthogonality, Orthogonal Complement and Orthonormal Families in a Real Inner Product Space §orthonormal, with $\langle e_{j},e_{j}\rangle=|e_{j}|^{2}=1$) give $\langle p_{m}^{*}(y),e_{j}\rangle=\sum_{i=1}^{m}y_{i}\langle e_{i},e_{j}\rangle$, which equals $y_{j}$ if $j\le m$ and $0$ if $j>m$. In particular, for $x\in X$ and $n\le m$ the first $n$ coordinates of $p_{m}^{*}(p_{m}(x))$ are $x_{1},\dots,x_{n}$, that is, $p_{n}(p_{m}^{*}(p_{m}(x)))=p_{n}(x)$. Also $(x+v)_{k}=x_{k}+v_{k}$ for $x,v\in X$, so $p_{m}(x+v)=p_{m}(x)+p_{m}(v)$.

(d) $\tau_{h}$ is Borel: for each $k$, the function $x\mapsto\langle\tau_{h}(x),e_{k}\rangle=x_{k}+h_{k}$ is the sum of the coordinate function $x\mapsto x_{k}$, which is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, and a constant, hence measurable by claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; so $\tau_{h}$ is measurable with respect to $\mathcal{B}(X)$ and $\mathcal{B}(X)$ by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §measurable, applied with $(\Omega,\mathcal{F})=(X,\mathcal{B}(X))$ and $Y=\tau_{h}$.

(e) $\rho_{h}$ and every $W_{v,m}$ ($v\in X$, $m\in\mathbb{N}$) are positive and Borel: $\ell_{h}$ and $\ell_{v,m}$ are Borel by The Paley-Wiener Functional of a Cameron-Martin Vector: Borel Measurability, Almost Everywhere and Mean-Square Convergence, Mean Zero and Variance the Cameron-Martin Square §borel, subtracting a constant preserves measurability by claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and (b) applies.

(f) For $m\in\mathbb{N}$, write $b=c^{(m)}$ and, for $a\in\mathbb{R}^{m}$, define $\theta_{a}:\mathbb{R}^{m}\to\mathbb{R}$ by $\theta_{a}(y)=\exp\bigl(y\cdot S_{b}(a)-\tfrac12|a|_{b}^{2}\bigr)$, with the dot product of Euclidean Space and Lebesgue Measure: Standing Notation §space. Then $\theta_{a}$ is positive and Borel: with $w=p_{m}^{*}(a)\in X$, (c) gives $w_{k}=a_{k}$ and $\langle p_{m}^{*}(y),e_{k}\rangle=y_{k}$ for $k\le m$, so $\ell_{w,m}(p_{m}^{*}(y))=\sum_{k=1}^{m}a_{k}y_{k}/c_{k}=y\cdot S_{b}(a)$; the map $p_{m}^{*}$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity and $\ell_{w,m}$ by The Paley-Wiener Functional of a Cameron-Martin Vector: Borel Measurability, Almost Everywhere and Mean-Square Convergence, Mean Zero and Variance the Cameron-Martin Square §borel, so $y\mapsto y\cdot S_{b}(a)$ is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, and (b) together with claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions applies. Moreover, if $v\in X$ and $a=p_{m}(v)$, then for every $x\in X$

$$\theta_{a}(p_{m}(x))=\exp\Bigl(\sum_{k=1}^{m}\frac{x_{k}v_{k}}{c_{k}}-\frac12\sum_{k=1}^{m}\frac{v_{k}^{2}}{c_{k}}\Bigr)=W_{v,m}(x).$$

Step 1 (Shift formula in $\mathbb{R}^{m}$). Let $m\in\mathbb{N}$, $b=c^{(m)}$, $a\in\mathbb{R}^{m}$, and let $g:\mathbb{R}^{m}\to[0,\infty)$ be Borel. We show

$$\int_{\mathbb{R}^{m}}g(y+a)\,\gamma_{b}(dy)=\int_{\mathbb{R}^{m}}g(y)\,\theta_{a}(y)\,\gamma_{b}(dy).$$

First, for $z\in\mathbb{R}^{m}$, expanding squares gives $|z-a|_{b}^{2}=\sum_{i=1}^{m}(z_{i}-a_{i})^{2}/b_{i}=|z|_{b}^{2}-2\,z\cdot S_{b}(a)+|a|_{b}^{2}$, so by the definition of $\rho_{b}$ and claim 1 of Basic Properties of the Exponential Function

$$\rho_{b}(z-a)=\exp\Bigl(-\tfrac12|z|_{b}^{2}-Z_{b}+z\cdot S_{b}(a)-\tfrac12|a|_{b}^{2}\Bigr)=\rho_{b}(z)\,\theta_{a}(z),$$

where $Z_{b}$ is the normalizing constant of The Diagonal Gaussian Density on Euclidean Space and Its Notation §density. By Diagonal Gaussian Measures on Euclidean Space §measure, $\gamma_{b}$ is the measure with density $\rho_{b}$ with respect to $\lambda_{m}$, where $\rho_{b}$ is positive and Borel by The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms §regularity. The function $y\mapsto g(y+a)$ is measurable by claim 2 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$, so claim 3 of Image Measures, Measures with Densities, and Change of Variables gives

$$\int_{\mathbb{R}^{m}}g(y+a)\,\gamma_{b}(dy)=\int_{\mathbb{R}^{m}}g(y+a)\,\rho_{b}(y)\,\lambda_{m}(dy).$$

Let $F(z)=g(z)\rho_{b}(z-a)$. The function $z\mapsto\rho_{b}(z-a)$ is measurable by claim 2 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$ (applied with $-a$ in place of $a$), so $F:\mathbb{R}^{m}\to[0,\infty)$ is measurable by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and $F(y+a)=g(y+a)\rho_{b}(y)$. Claim 2 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$ gives $\int F(y+a)\,\lambda_{m}(dy)=\int F\,d\lambda_{m}$, and $F=(g\theta_{a})\rho_{b}$ pointwise by the identity above. The function $g\theta_{a}$ is nonnegative and measurable by (f) and claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, so claim 3 of Image Measures, Measures with Densities, and Change of Variables once more gives $\int(g\theta_{a})\rho_{b}\,d\lambda_{m}=\int g\theta_{a}\,d\gamma_{b}$. Chaining the equalities proves the formula.

Step 2 (Shift formula on cylinder functions). Let $v\in X$, $m\in\mathbb{N}$, $b=c^{(m)}$, and let $g:\mathbb{R}^{m}\to[0,\infty)$ be Borel. We show

$$\int_{X}g(p_{m}(x+v))\,\gamma_{c}(dx)=\int_{X}g(p_{m}(x))\,W_{v,m}(x)\,\gamma_{c}(dx).$$

Let $a=p_{m}(v)$ and $G(y)=g(y+a)$, which is Borel by claim 2 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$; by (c), $g(p_{m}(x+v))=G(p_{m}(x))$. The map $p_{m}$ is Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity, and $(p_{m})_{\#}\gamma_{c}=\gamma_{b}$ by Diagonal Gaussian Measures on a Hilbert Space §measure, the push-forward being the image measure of claim 1 of Image Measures, Measures with Densities, and Change of Variables by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward. Claim 2 of Image Measures, Measures with Densities, and Change of Variables (with $T=p_{m}$), then Step 1, then claim 2 of Image Measures, Measures with Densities, and Change of Variables again, applied to the nonnegative Borel function $g\theta_{a}$ (Borel by (f) and claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), give

$$\int_{X}G\circ p_{m}\,d\gamma_{c}=\int_{\mathbb{R}^{m}}G\,d\gamma_{b}=\int_{\mathbb{R}^{m}}g\,\theta_{a}\,d\gamma_{b}=\int_{X}g(p_{m}(x))\,\theta_{a}(p_{m}(x))\,\gamma_{c}(dx),$$

and $\theta_{a}(p_{m}(x))=W_{v,m}(x)$ by (f).

Two consequences. (2a) Taking $g$ the constant $1$, the left side is $\int_{X}\mathbf{1}_{X}\,d\gamma_{c}=\gamma_{c}(X)=1$ by The Integral of an Indicator Function is the Measure of the Set, so $\int_{X}W_{v,m}\,d\gamma_{c}=1$ for all $v\in X$ and $m\in\mathbb{N}$.

(2b) Let $n\le m$, $B\in\mathcal{B}(\mathbb{R}^{n})$ and $A=p_{n}^{-1}(B)$, a Borel subset of $X$ since $p_{n}$ is Borel. Put $B'=(p_{m}^{*})^{-1}(p_{n}^{-1}(B))$, which belongs to $\mathcal{B}(\mathbb{R}^{m})$ because $p_{n}$ and $p_{m}^{*}$ are Borel by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity. By (c), $p_{m}(x)\in B'$ if and only if $p_{n}(x)\in B$, that is, $\mathbf{1}_{B'}(p_{m}(x))=\mathbf{1}_{A}(x)$ for every $x\in X$; in particular $\mathbf{1}_{B'}(p_{m}(x+h))=\mathbf{1}_{A}(x+h)=\mathbf{1}_{\tau_{h}^{-1}(A)}(x)$, where $\tau_{h}^{-1}(A)$ is Borel by (d). Applying the formula with $v=h$ and $g=\mathbf{1}_{B'}$ (Borel by claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) and using The Integral of an Indicator Function is the Measure of the Set on the left,

$$\gamma_{c}\bigl(\tau_{h}^{-1}(A)\bigr)=\int_{X}\mathbf{1}_{A}\,W_{m}\,d\gamma_{c}\qquad(n\le m).$$

Step 3 (Mean-square convergence of $W_{m}$ to $\rho_{h}$). Let $m\le k$ and put $v=h+p_{m}^{*}(p_{m}(h))\in X$. By (c), $v_{j}=2h_{j}$ for $j\le m$ and $v_{j}=h_{j}$ for $j>m$. Hence $\ell_{v,k}=\ell_{h,k}+\ell_{h,m}$ and $s_{v,k}=s_{k}+3s_{m}$, so

$$\ell_{v,k}-\tfrac12 s_{v,k}=\bigl(\ell_{h,m}-\tfrac12 s_{m}\bigr)+\bigl(\ell_{h,k}-\tfrac12 s_{k}\bigr)-s_{m},$$

and claim 1 of Basic Properties of the Exponential Function gives $W_{m}W_{k}=\exp(s_{m})\,W_{v,k}$ pointwise. By (2a) and Linearity and Monotonicity of the Lebesgue Integral §nonnegative,

$$\int_{X}W_{m}W_{k}\,d\gamma_{c}=\exp(s_{m})\qquad(m\le k);$$

in particular $\int_{X}W_{m}^{2}\,d\gamma_{c}=\exp(s_{m})$. For $m\le k$ we have pointwise $(W_{k}-W_{m})^{2}+2W_{m}W_{k}=W_{k}^{2}+W_{m}^{2}$, all terms being nonnegative and measurable by (e) and claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; additivity in Linearity and Monotonicity of the Lebesgue Integral §nonnegative gives $\int(W_{k}-W_{m})^{2}\,d\gamma_{c}+2\exp(s_{m})=\exp(s_{k})+\exp(s_{m})$; the right side is finite, hence so is $\int(W_{k}-W_{m})^{2}\,d\gamma_{c}$, and subtracting the real number $2\exp(s_{m})$ from both sides is permitted, so

$$\int_{X}(W_{k}-W_{m})^{2}\,d\gamma_{c}=\exp(s_{k})-\exp(s_{m})\qquad(m\le k).$$

Let $C$ be the set of $x\in X$ for which $(\ell_{h,n}(x))_{n\in\mathbb{N}}$ converges. By The Paley-Wiener Functional of a Cameron-Martin Vector: Borel Measurability, Almost Everywhere and Mean-Square Convergence, Mean Zero and Variance the Cameron-Martin Square §borel and The Paley-Wiener Functional of a Cameron-Martin Vector: Borel Measurability, Almost Everywhere and Mean-Square Convergence, Mean Zero and Variance the Cameron-Martin Square §almost-everywhere, $C$ is Borel with $\gamma_{c}(C)=1$, so $X\setminus C$ is a Borel set with $\gamma_{c}(X\setminus C)=0$, hence null. For $x\in C$, $\ell_{h,k}(x)\to\ell_{h}(x)$ by The Paley-Wiener Functional of a Vector Relative to a Variance Sequence §functional, and $s_{k}\to|h|_{c}^{2}$ because $|h|_{c}^{2}$ is the sum of the series of The Cameron-Martin Space of a Diagonal Gaussian Measure on a Hilbert Space §square; by (b), $W_{k}(x)\to\rho_{h}(x)$ as $k\to\infty$, and $\exp(s_{k})\to\exp(|h|_{c}^{2})$. Since the terms $h_{j}^{2}/c_{j}$ are nonnegative, $s_{k}\le|h|_{c}^{2}$ for every $k$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates, so by claim 4 of Basic Properties of the Exponential Function

$$\delta_{m}:=\exp(|h|_{c}^{2})-\exp(s_{m})\ge0,\qquad\delta_{m}\to0\ (m\to\infty).$$

Fix $m$, and for $k\in\mathbb{N}$ let $f_{k}=(W_{\max(k,m)}-W_{m})^{2}$, a measurable function $X\to[0,\infty)$. Then $\int f_{k}\,d\gamma_{c}=\exp(s_{\max(k,m)})-\exp(s_{m})\le\delta_{m}$ for every $k$ (claim 4 of Basic Properties of the Exponential Function), so every infimum $\inf_{j\ge K}\int f_{j}\,d\gamma_{c}$ is at most $\delta_{m}$ and $\liminf_{k}\int f_{k}\,d\gamma_{c}\le\delta_{m}$. For $x\in C$ the real sequence $(f_{k}(x))_{k}$ converges to $L:=(\rho_{h}(x)-W_{m}(x))^{2}$, and its liminf $\sup_{K}\inf_{j\ge K}f_{j}(x)$, in the sense of Fatou's Lemma, equals $L$. Indeed, let $\varepsilon>0$ and choose $K_{0}$ with $|f_{j}(x)-L|<\varepsilon$ for all $j\ge K_{0}$. For every $K$, the index $j=\max(K,K_{0})$ satisfies $j\ge K$ and $f_{j}(x)<L+\varepsilon$, so $\inf_{j\ge K}f_{j}(x)\le L+\varepsilon$; and for $K\ge K_{0}$ every $f_{j}(x)$ with $j\ge K$ exceeds $L-\varepsilon$, so $\inf_{j\ge K}f_{j}(x)\ge L-\varepsilon$. Hence $L-\varepsilon\le\sup_{K}\inf_{j\ge K}f_{j}(x)\le L+\varepsilon$, and since $\varepsilon>0$ is arbitrary, $\sup_{K}\inf_{j\ge K}f_{j}(x)=L$. Thus the measurable functions $\liminf_{k}f_{k}$ (measurable by Fatou's Lemma) and $(\rho_{h}-W_{m})^{2}$ (measurable by (e) and claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) agree outside the null set $X\setminus C$, and The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison together with Fatou's Lemma gives

$$\int_{X}(\rho_{h}-W_{m})^{2}\,d\gamma_{c}=\int_{X}\Bigl(\liminf_{k}f_{k}\Bigr)\,d\gamma_{c}\le\liminf_{k}\int_{X}f_{k}\,d\gamma_{c}\le\delta_{m}.$$

Now let $\varepsilon>0$ be real. For every real $u$, $0\le(|u|-\varepsilon)^{2}$ gives $|u|\le\tfrac{\varepsilon}{2}+\tfrac{u^{2}}{2\varepsilon}$. The function $|\rho_{h}-W_{m}|$ is measurable by claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and by Linearity and Monotonicity of the Lebesgue Integral §nonnegative and The Integral of an Indicator Function is the Measure of the Set (the constant $\varepsilon/2$ being $\tfrac{\varepsilon}{2}\mathbf{1}_{X}$, with $\gamma_{c}(X)=1$)

$$\eta_{m}:=\int_{X}|\rho_{h}-W_{m}|\,d\gamma_{c}\le\frac{\varepsilon}{2}+\frac{1}{2\varepsilon}\int_{X}(\rho_{h}-W_{m})^{2}\,d\gamma_{c}\le\frac{\varepsilon}{2}+\frac{\delta_{m}}{2\varepsilon}.$$

Taking $\varepsilon=1$ gives $\eta_{m}\le\tfrac12+\tfrac12\delta_{m}<\infty$, so each $\eta_{m}$ is a real number. For general $\varepsilon>0$, choosing $M$ with $\delta_{m}\le\varepsilon^{2}$ for all $m\ge M$ (possible since $\delta_{m}\to0$) gives $\eta_{m}\le\varepsilon$ for $m\ge M$; hence $\eta_{m}\to0$. Finally, $\rho_{h}\le W_{m}+|\rho_{h}-W_{m}|$ pointwise, so Linearity and Monotonicity of the Lebesgue Integral §nonnegative and (2a) give $\int\rho_{h}\,d\gamma_{c}\le1+\eta_{m}<\infty$; as $\rho_{h}=|\rho_{h}|$ is measurable, $\rho_{h}$ is integrable by Measure Spaces and the Lebesgue Integral: Standing Notation §integral. Likewise each $W_{m}$ is integrable by (2a).

Step 4 (Claims 1 and 2). Let $n\in\mathbb{N}$, $B\in\mathcal{B}(\mathbb{R}^{n})$, $A=p_{n}^{-1}(B)$ and $m\ge n$. The functions $\mathbf{1}_{A}W_{m}$ and $\mathbf{1}_{A}\rho_{h}$ are measurable (claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), nonnegative, and bounded by the integrable functions $W_{m}$ and $\rho_{h}$, so they are integrable by Linearity and Monotonicity of the Lebesgue Integral §nonnegative and Measure Spaces and the Lebesgue Integral: Standing Notation §integral; for a nonnegative integrable function the integral of Integrable Function and the Lebesgue Integral equals its integral as a nonnegative function, since its negative part vanishes. By Linearity and Monotonicity of the Lebesgue Integral §integrable and the monotonicity in Linearity and Monotonicity of the Lebesgue Integral §nonnegative,

$$\Bigl|\int_{X}\mathbf{1}_{A}W_{m}\,d\gamma_{c}-\int_{X}\mathbf{1}_{A}\rho_{h}\,d\gamma_{c}\Bigr|\le\int_{X}\mathbf{1}_{A}|W_{m}-\rho_{h}|\,d\gamma_{c}\le\eta_{m}.$$

By (2b) the first integral equals $\gamma_{c}(\tau_{h}^{-1}(A))$ for every $m\ge n$; letting $m\to\infty$ and using $\eta_{m}\to0$,

$$\gamma_{c}\bigl(\tau_{h}^{-1}(p_{n}^{-1}(B))\bigr)=\int_{X}\mathbf{1}_{p_{n}^{-1}(B)}\,\rho_{h}\,d\gamma_{c}\qquad(n\in\mathbb{N},\ B\in\mathcal{B}(\mathbb{R}^{n})).\tag{$*$}$$

Taking $n=1$ and $B=\mathbb{R}^{1}$, so that $p_{1}^{-1}(B)=X=\tau_{h}^{-1}(X)$ and $\mathbf{1}_{X}\rho_{h}=\rho_{h}$, gives $\int_{X}\rho_{h}\,d\gamma_{c}=\gamma_{c}(X)=1$. Together with (d) and (e), this proves claim 1.

Since $\rho_{h}:X\to[0,\infty)$ is measurable, claim 3 of Image Measures, Measures with Densities, and Change of Variables provides the measure $\nu$ on $(X,\mathcal{B}(X))$ with density $\rho_{h}$ with respect to $\gamma_{c}$, $\nu(A)=\int_{X}\mathbf{1}_{A}\rho_{h}\,d\gamma_{c}$; it is a Borel measure on $(X,d)$ by Borel Measure on a Metric Space, and $\nu(X)=\int_{X}\rho_{h}\,d\gamma_{c}=1$. Let $\mu'=(\tau_{h})_{\#}\gamma_{c}$, a Borel measure on $(X,d)$ with $\mu'(X)=\gamma_{c}(X)=1$ by (d) and claim 1 of Image Measures, Measures with Densities, and Change of Variables. For every $n\in\mathbb{N}$ and $B\in\mathcal{B}(\mathbb{R}^{n})$, by the definition of image measures and $(*)$,

$$\bigl((p_{n})_{\#}\mu'\bigr)(B)=\mu'(p_{n}^{-1}(B))=\gamma_{c}\bigl(\tau_{h}^{-1}(p_{n}^{-1}(B))\bigr)=\nu(p_{n}^{-1}(B))=\bigl((p_{n})_{\#}\nu\bigr)(B).$$

So $(p_{n})_{\#}\mu'=(p_{n})_{\#}\nu$ for every $n$, and $\mu'(X)=\nu(X)=1<\infty$; by Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §determination, $\mu'=\nu$. That is, $((\tau_{h})_{\#}\gamma_{c})(B)=\nu(B)=\int_{X}\mathbf{1}_{B}\rho_{h}\,d\gamma_{c}$ for every Borel $B\subseteq X$, and $(\tau_{h})_{\#}\gamma_{c}$ is the measure with density $\rho_{h}$ with respect to $\gamma_{c}$. This proves claim 2.

Step 5 (Claim 3). Let $f:X\to\mathbb{R}$ be Borel, so that $f\circ\tau_{h}$, the function $x\mapsto f(x+h)$, is Borel by (d) and claim 4 of Borel Measurability and Bounded Integration on a Metric Space. If $f$ is nonnegative, claim 2 of Image Measures, Measures with Densities, and Change of Variables with $T=\tau_{h}$, then $\mu'=\nu$ from Step 4, then claim 3 of Image Measures, Measures with Densities, and Change of Variables give, in $[0,\infty]$,

$$\int_{X}f(x+h)\,\gamma_{c}(dx)=\int_{X}f\,d\mu'=\int_{X}f\,d\nu=\int_{X}f(x)\,\rho_{h}(x)\,\gamma_{c}(dx).$$

For general Borel $f$, claim 2 of Image Measures, Measures with Densities, and Change of Variables says that $f\circ\tau_{h}$ is integrable with respect to $\gamma_{c}$ if and only if $f$ is integrable with respect to $\mu'=\nu$, and claim 3 of the same lemma says that this holds if and only if $f\rho_{h}$ is integrable with respect to $\gamma_{c}$; in that case the same two claims give the three equalities displayed above in $\mathbb{R}$. This proves claim 3.