Proof of The Mehler Law Flow: Duality, Finite Second Moments, Invariance of the Gaussian Measure, Generation by the Noise Ornstein-Uhlenbeck Functional, and Decrease of Relative Entropy

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

Throughout, $\gamma_{c}\in\mathcal{P}_{2}(X)$ by Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §moment. Probability measures are finite, hence $\sigma$-finite, so Existence and Uniqueness of the Product Measure and Tonelli and Fubini Theorems apply to the product of two of them, and such a product is again a probability measure, its value at $X\times X$ being $1\cdot1$ by Existence and Uniqueness of the Product Measure. By The Mehler Law Flow of a Probability Measure on a Hilbert Space the Borel $\sigma$-algebra of $X\times X$ is $\mathcal{B}(X)\otimes\mathcal{B}(X)$, and for $\nu\in\mathcal{P}(X)$ and real $r\ge0$ we have $\nu P_{r}=(M_{r})_{\#}(\nu\otimes\gamma_{c})$ by The Mehler Law Flow of a Probability Measure on a Hilbert Space §law-flow, the Mehler map $M_{r}$ being Borel by The Mehler Maps and the Mehler Semigroup with Noise Weights on Continuous Cylindrical Functions of Polynomial Growth §map. For a Borel function $u:X\to\mathbb{R}$, the functions $(x,y)\mapsto u(x)$ and $(x,y)\mapsto u(y)$ on $X\times X$ are measurable with respect to $\mathcal{B}(X)\otimes\mathcal{B}(X)$: the preimages of a Borel set $A\subseteq X$ under the two coordinate maps are the measurable rectangles $A\times X$ and $X\times A$, so these maps are measurable by Product Sigma-Algebra, and the composites are measurable by claim 4 of Borel Measurability and Bounded Integration on a Metric Space. Every $F\in\mathcal{F}C_{\mathrm{pol}}(X)$ is continuous, hence Borel, by The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §class; and if $(n,\psi,B,q)$ is a representation of $F$ and $r\ge0$ is real, then by The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §cylindrical the function $P_{r}F$ lies in $\mathcal{F}C_{\mathrm{pol}}(X)$ with a representation $(n,\psi_{r},B',q)$ of the same growth exponent $q$, where $B'$ does not depend on $r$, and $|P_{r}F(x)|=|\psi_{r}(p_{n}(x))|\le B'(1+\lVert p_{n}(x)\rVert^{q})$ for every $x\in X$. Finally $\lVert p_{n}(z)\rVert\le|z|$ for $z\in X$: by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square, $\lVert p_{n}(z)\rVert^{2}=\sum_{k=1}^{n}z_{k}^{2}$, which is at most $\sum_{k=1}^{\infty}z_{k}^{2}\le|z|^{2}$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates and Orthonormal Expansions in a Real Hilbert Space §bessel, and square roots preserve this inequality by claim 5 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities.

Step 1 (Second moments). Let $\nu\in\mathcal{P}_{2}(X)$ and let $r\ge0$ be real. By The Mehler Maps and the Mehler Semigroup with Noise Weights on Continuous Cylindrical Functions of Polynomial Growth §map, $|M_{r}(x,y)|\le|x|+|y|$, hence $|M_{r}(x,y)|^{2}\le2|x|^{2}+2|y|^{2}$, for all $x,y\in X$. The function $z\mapsto|z|^{2}$ is Borel and nonnegative by The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment. By claim 2 of Image Measures, Measures with Densities, and Change of Variables, the monotonicity and additivity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative, and Tonelli's theorem of Tonelli and Fubini Theorems,

$$M_{2}(\nu P_{r})=\int_{X\times X}|M_{r}(x,y)|^{2}\,(\nu\otimes\gamma_{c})(dx,dy)\le\int_{X}\int_{X}\bigl(2|x|^{2}+2|y|^{2}\bigr)\,\gamma_{c}(dy)\,\nu(dx)=2M_{2}(\nu)+2M_{2}(\gamma_{c})<\infty,$$

where $M_{2}$ is the second moment of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §moment (a number attached to a measure, not one of the Mehler maps $M_{r}$) and the inner integral equals $2|x|^{2}+2M_{2}(\gamma_{c})$ because $\gamma_{c}(X)=1$. Hence $\nu P_{r}\in\mathcal{P}_{2}(X)$ by The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space; in particular $\mu P_{t}\in\mathcal{P}_{2}(X)$. The same computation shows that the function $w(x,y)=2+2|x|^{2}+2|y|^{2}$ on $X\times X$ is measurable, nonnegative and integrable with respect to $\nu\otimes\gamma_{c}$, with integral $2+2M_{2}(\nu)+2M_{2}(\gamma_{c})$.

Step 2 (Duality for every measure with finite second moment). We show: for every $\nu\in\mathcal{P}_{2}(X)$, every real $r\ge0$ and every $F\in\mathcal{F}C_{\mathrm{pol}}(X)$ having a representation $(n,\psi,B,q)$ with $q\le2$, the function $F$ is integrable with respect to $\nu P_{r}$, the function $P_{r}F$ is integrable with respect to $\nu$, and

$$\int_{X}F\,d(\nu P_{r})=\int_{X}P_{r}F\,d\nu .\tag{1}$$

Claim 2 is the case $\nu=\mu$, $r=t$. Since $q\in\{0,1,2\}$, we have $u^{q}\le1+u^{2}$ for real $u\ge0$, so $|F(z)|\le B(1+\lVert p_{n}(z)\rVert^{q})\le B(2+|z|^{2})$ for $z\in X$, and with Step 1, $|F(M_{r}(x,y))|\le B(2+2|x|^{2}+2|y|^{2})=B\,w(x,y)$. The function $F\circ M_{r}$ is measurable with respect to $\mathcal{B}(X)\otimes\mathcal{B}(X)$ by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, and $\int|F\circ M_{r}|\,d(\nu\otimes\gamma_{c})\le B\int w\,d(\nu\otimes\gamma_{c})<\infty$ by Linearity and Monotonicity of the Lebesgue Integral §nonnegative; so $F\circ M_{r}$ is integrable with respect to $\nu\otimes\gamma_{c}$. By claim 2 of Image Measures, Measures with Densities, and Change of Variables, $F$ is integrable with respect to $\nu P_{r}=(M_{r})_{\#}(\nu\otimes\gamma_{c})$ and $\int_{X}F\,d(\nu P_{r})=\int_{X\times X}F\circ M_{r}\,d(\nu\otimes\gamma_{c})$. By Fubini's theorem of Tonelli and Fubini Theorems there is a $\nu$-null set $N\in\mathcal{B}(X)$ such that the function $\tilde{h}$ equal to $\int_{X}F(M_{r}(x,y))\,\gamma_{c}(dy)$ for $x\notin N$ and to $0$ on $N$ is integrable with respect to $\nu$ with $\int_{X}\tilde{h}\,d\nu=\int_{X\times X}F\circ M_{r}\,d(\nu\otimes\gamma_{c})$. By The Mehler Maps and the Mehler Semigroup with Noise Weights on Continuous Cylindrical Functions of Polynomial Growth §semigroup, $\tilde{h}(x)=P_{r}F(x)$ for $x\notin N$, and $P_{r}F$ is Borel, lying in $\mathcal{F}C_{\mathrm{pol}}(X)$; so by the third part of The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison, $P_{r}F$ is integrable with respect to $\nu$ with $\int_{X}P_{r}F\,d\nu=\int_{X}\tilde{h}\,d\nu$. This proves (1).

Step 3 (Bounded cylindrical functions determine a measure). We show: if $\nu_{1},\nu_{2}\in\mathcal{P}(X)$ satisfy $\int_{X}F\,d\nu_{1}=\int_{X}F\,d\nu_{2}$ for every $F\in\mathcal{F}C_{\mathrm{pol}}(X)$ having a representation with growth exponent $0$, then $\nu_{1}=\nu_{2}$. (Such an $F$, with representation $(n,\psi,B,0)$, satisfies $|F|\le2B$, so it is integrable with respect to every probability measure by claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space.) Let $f:X\to\mathbb{R}$ be bounded and Lipschitz, with $|f(x)|\le K$ and $|f(x)-f(x')|\le\Lambda|x-x'|$ for $x,x'\in X$. For $n\in\mathbb{N}$ let $\psi^{(n)}=f\circ p_{n}^{*}$, with the synthesis map $p_{n}^{*}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates. For $y,y'\in\mathbb{R}^{n}$, $p_{n}^{*}(y)-p_{n}^{*}(y')=\sum_{k=1}^{n}(y_{k}-y'_{k})e_{k}$ is the sum of the series $\sum_{k}d_{k}e_{k}$ with $d_{k}=y_{k}-y'_{k}$ for $k\le n$ and $d_{k}=0$ for $k>n$ (its partial sums are eventually constant), so by Orthonormal Expansions in a Real Hilbert Space §riesz-fischer and Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square, $|p_{n}^{*}(y)-p_{n}^{*}(y')|^{2}=\sum_{k=1}^{n}(y_{k}-y'_{k})^{2}=\lVert y-y'\rVert^{2}$. Hence $|\psi^{(n)}(y)-\psi^{(n)}(y')|\le\Lambda\lVert y-y'\rVert$, so $\psi^{(n)}$ is continuous, and $|\psi^{(n)}(y)|\le K\le K(1+\lVert y\rVert^{0})$; thus $F_{n}=\psi^{(n)}\circ p_{n}$ lies in $\mathcal{F}C_{\mathrm{pol}}(X)$ with representation $(n,\psi^{(n)},K,0)$ by Continuous Cylindrical Functions of Polynomial Growth on a Hilbert Space §class. Now $F_{n}(x)=f(p_{n}^{*}(p_{n}(x)))$ with $p_{n}^{*}(p_{n}(x))=\sum_{k=1}^{n}x_{k}e_{k}$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, and these partial sums converge to $x$ by Orthonormal Expansions in a Real Hilbert Space §expansion; so $|F_{n}(x)-f(x)|\le\Lambda|p_{n}^{*}(p_{n}(x))-x|\to0$ for every $x\in X$. Since $|F_{n}|\le K$, the constant $K$ is integrable with respect to $\nu_{i}$ by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space, and $f$ is Borel (being Lipschitz, hence continuous) by claim 3 of that lemma, Dominated Convergence Theorem gives $\int_{X}F_{n}\,d\nu_{i}\to\int_{X}f\,d\nu_{i}$ for $i=1,2$. As $\int_{X}F_{n}\,d\nu_{1}=\int_{X}F_{n}\,d\nu_{2}$ for every $n$, we get $\int_{X}f\,d\nu_{1}=\int_{X}f\,d\nu_{2}$ for every bounded Lipschitz $f$, and $\nu_{1}=\nu_{2}$ by claim 1 of Lipschitz Test Functions Determine a Finite Borel Measure, and Uniqueness of Weak Limits ($X$ is nonempty and $\nu_{1},\nu_{2}$ are finite).

Step 4 (Claim 1). We have $\mu P_{t}\in\mathcal{P}_{2}(X)$ by Step 1. Let $F\in\mathcal{F}C_{\mathrm{pol}}(X)$ have a representation $(n,\psi,B,0)$. Since $P_{0}F=F$ by The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §cylindrical, (1) with $\nu=\mu$, $r=0$ gives $\int_{X}F\,d(\mu P_{0})=\int_{X}F\,d\mu$; by Step 3, $\mu P_{0}=\mu$. Next, $\mu P_{s}\in\mathcal{P}_{2}(X)$ by Step 1, and $P_{t}F$ has a representation with growth exponent $0$, as recalled at the start. Applying (1) three times, with $(\nu,r,F)$ equal to $(\mu P_{s},t,F)$, $(\mu,s,P_{t}F)$ and $(\mu,s+t,F)$, together with The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §semigroup,

$$\int_{X}F\,d\bigl((\mu P_{s})P_{t}\bigr)=\int_{X}P_{t}F\,d(\mu P_{s})=\int_{X}P_{s}(P_{t}F)\,d\mu=\int_{X}P_{s+t}F\,d\mu=\int_{X}F\,d(\mu P_{s+t}),$$

so $(\mu P_{s})P_{t}=\mu P_{s+t}$ by Step 3. Finally (1) with $\nu=\gamma_{c}$ and The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §invariance give $\int_{X}F\,d(\gamma_{c}P_{t})=\int_{X}P_{t}F\,d\gamma_{c}=\int_{X}F\,d\gamma_{c}$, so $\gamma_{c}P_{t}=\gamma_{c}$ by Step 3.

Step 5 (Claim 3). Let $\nu\in\mathcal{P}_{2}(X)$, $n\in\mathbb{N}$, $g\in C^{2}_{b}(\mathbb{R}^{n})$, and $F=g\circ p_{n}$, which lies in $\mathcal{F}C^{2}_{b}(X)$ by Bounded C^2 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical. By The Ornstein-Uhlenbeck Operator with Noise Weights on Bounded C^2 Cylindrical Functions §second-derivatives, for $k\le n$ we have $\partial_{k}g\in C^{1}_{b}(\mathbb{R}^{n})$ and $\partial_{k}F=(\partial_{k}g)\circ p_{n}$, so $(n,\partial_{k}g)$ is a representation of $\partial_{k}F\in\mathcal{F}C^{1}_{b}(X)$ and Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial gives $\partial_{k}\partial_{k}F=(\partial_{k}\partial_{k}g)\circ p_{n}$. Since $x_{k}$ is the $k$-th component of $p_{n}(x)$, The Ornstein-Uhlenbeck Operator with Noise Weights on Bounded C^2 Cylindrical Functions §operator gives, for every $x\in X$,

$$L^{a}F(x)=-\sum_{k=1}^{n}a_{k}\,u_{k}(x),\qquad u_{k}(x)=\frac{x_{k}}{c_{k}}\,\partial_{k}g(p_{n}(x))-\partial_{k}\partial_{k}g(p_{n}(x)).\tag{2}$$

By The Noise Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates §functional, each $u_{k}$ is integrable with respect to $\nu$ and $L^{a}_{\nu}(g)=\sum_{k=1}^{n}a_{k}\int_{X}u_{k}\,d\nu$. By Linearity and Monotonicity of the Lebesgue Integral §integrable, $L^{a}F$ is integrable with respect to $\nu$ and $\int_{X}L^{a}F\,d\nu=-\sum_{k=1}^{n}a_{k}\int_{X}u_{k}\,d\nu=-L^{a}_{\nu}(g)$.

Step 6 (The growth exponent of $L^{a}F$). With $n$, $g$ and $F=g\circ p_{n}$ as in Step 5, (2) gives $L^{a}F=\ell\circ p_{n}$ with

$$\ell(y)=\sum_{k=1}^{n}a_{k}\Bigl(\partial_{k}\partial_{k}g(y)-\frac{y_{k}}{c_{k}}\,\partial_{k}g(y)\Bigr)\qquad(y\in\mathbb{R}^{n}).$$

Since $g$ is of class $C^{2}$, it is of class $C^{1}$ and each $\partial_{k}g$ is of class $C^{1}$ by clause 2 of C^k Maps on a Euclidean Open Set, so $\partial_{k}g$ and $\partial_{k}\partial_{k}g$ are continuous by clause 1 there; the map $y\mapsto y_{k}$ is continuous since $|y_{k}-y'_{k}|\le\lVert y-y'\rVert$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square; hence $\ell$ is continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set. By Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded there is a real $C\ge0$ with $|\partial_{k}g|\le C$ and $|\partial_{k}\partial_{k}g|\le C$ on $\mathbb{R}^{n}$ for $k\le n$, and $|y_{k}|\le\lVert y\rVert$; so $|\ell(y)|\le B_{1}(1+\lVert y\rVert)$ with $B_{1}=C\sum_{k=1}^{n}a_{k}(1+c_{k}^{-1})$. Thus $(n,\ell,B_{1},1)$ is a representation of $L^{a}F\in\mathcal{F}C_{\mathrm{pol}}(X)$, of growth exponent $1$.

Step 7 (Claim 4). Let $n$, $g$ and $F=g\circ p_{n}$ be as in Step 5. By Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded there is $C_{0}\ge0$ with $|g|\le C_{0}$, and $g$ is continuous by Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, so $(n,g,C_{0},0)$ is a representation of $F$. Step 1 and (1), with $\nu=\mu$, give $\Phi(r)=\int_{X}P_{r}F\,d\mu$ for every real $r\ge0$, each $P_{r}F$ being $\mu$-integrable. By Step 6 and The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §cylindrical there is a real $B'\ge0$ with

$$|P_{r}(L^{a}F)(x)|\le B'(1+\lVert p_{n}(x)\rVert)\le B'(1+|x|)\qquad(r\ge0,\ x\in X).\tag{3}$$

The function $x\mapsto B'(1+|x|)$ is continuous, hence Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space, and integrable with respect to $\mu$, since $0\le1+|x|\le2+|x|^{2}$ and $\int_{X}(2+|x|^{2})\,d\mu=2+M_{2}(\mu)<\infty$, both by Linearity and Monotonicity of the Lebesgue Integral §nonnegative (the constant $2$ having integral $2\mu(X)=2$ by claim 6 of Borel Measurability and Bounded Integration on a Metric Space).

Fix $x\in X$ and put $\varphi_{x}(r)=P_{r}F(x)$ for real $r\ge0$. By The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §continuity, $\varphi_{x}$ is continuous on $[0,\infty)$. For real $r>0$, the limit of The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §generator (applied at time $r$) is taken over all real $h\ne0$ with $|h|<r$ among others, so $\varphi_{x}$ is differentiable at the interior point $r$ with derivative $P_{r}(L^{a}F)(x)$. Let $h\ne0$ be real with $t+h\ge0$, and let $\alpha$ and $\beta$ be the smaller and the larger of $t$ and $t+h$. The restriction of $\varphi_{x}$ to $[\alpha,\beta]$ is continuous, and differentiable at every point of the open interval between $\alpha$ and $\beta$ with the same derivative (for such a point the difference quotients of the restriction are those of $\varphi_{x}$ for small increments); so Mean Value Theorem on a Closed Real Interval gives $\xi$ with $\alpha<\xi<\beta$, in particular $\xi>0$, and

$$\frac{P_{t+h}F(x)-P_{t}F(x)}{h}=P_{\xi}(L^{a}F)(x),\qquad\text{hence}\qquad\Bigl|\frac{P_{t+h}F(x)-P_{t}F(x)}{h}\Bigr|\le B'(1+|x|)\tag{4}$$

by (3). Now let $(h_{j})_{j\in\mathbb{N}}$ be a sequence of real numbers with $h_{j}\ne0$, $t+h_{j}\ge0$ and $h_{j}\to0$, and put $D_{j}(x)=(P_{t+h_{j}}F(x)-P_{t}F(x))/h_{j}$. Each $D_{j}$ is Borel by Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; for every $x$, $D_{j}(x)\to P_{t}(L^{a}F)(x)$ by The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §generator (given $\varepsilon>0$ and the corresponding $\delta>0$ there, $|h_{j}|<\delta$ for all large $j$); and $|D_{j}(x)|\le B'(1+|x|)$ by (4). By Dominated Convergence Theorem and the linearity of Linearity and Monotonicity of the Lebesgue Integral §integrable,

$$\frac{\Phi(t+h_{j})-\Phi(t)}{h_{j}}=\int_{X}D_{j}\,d\mu\longrightarrow\int_{X}P_{t}(L^{a}F)\,d\mu=\int_{X}L^{a}F\,d(\mu P_{t})=-L^{a}_{\mu P_{t}}(g),$$

where the first equality on the right is (1) with $\nu=\mu$, $r=t$, applied to $L^{a}F$, which has growth exponent $1\le2$ by Step 6, and the last is Step 5 with $\nu=\mu P_{t}\in\mathcal{P}_{2}(X)$. Finally, if the difference quotients $(\Phi(t+h)-\Phi(t))/h$ did not converge to $-L^{a}_{\mu P_{t}}(g)$ as $h\to0$ over real $h\ne0$ with $t+h\ge0$, there would be $\varepsilon>0$ and, for every $j\in\mathbb{N}$, such an $h_{j}$ with $|h_{j}|<1/j$ and $|(\Phi(t+h_{j})-\Phi(t))/h_{j}+L^{a}_{\mu P_{t}}(g)|\ge\varepsilon$, contradicting what was just shown for the sequence $(h_{j})$. This proves claim 4.

Step 8 (Claim 5). Suppose $\mu$ has finite relative entropy with respect to $\gamma_{c}$: by Relative Entropy of Probability Measures §relative-entropy there is a density $f$ of $\mu$ with respect to $\gamma_{c}$, in the sense of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities (Borel, nonnegative, with $\mu(A)=\int_{X}\mathbf{1}_{A}f\,d\gamma_{c}$ for every Borel $A$), such that $\phi\circ f$ is integrable with respect to $\gamma_{c}$, where $\phi$ is the function of The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm, and $H(\mu\,|\,\gamma_{c})=\int_{X}\phi\circ f\,d\gamma_{c}$. Put $\tilde{f}(x,y)=f(x)$ on $X\times X$, a nonnegative measurable function. Let $E\in\mathcal{B}(X)\otimes\mathcal{B}(X)$. By Tonelli's theorem of Tonelli and Fubini Theorems, the function $x\mapsto\int_{X}\mathbf{1}_{E}(x,y)\,\gamma_{c}(dy)$ is Borel with values in $[0,1]$; by claim 3 of Image Measures, Measures with Densities, and Change of Variables, the measure with density $f$ with respect to $\gamma_{c}$ being $\mu$, it may be integrated against $\mu$ by multiplying by $f$ and integrating against $\gamma_{c}$; and $f(x)$ may be taken inside the inner integral by Linearity and Monotonicity of the Lebesgue Integral §nonnegative. Hence, by Tonelli's theorem twice,

$$(\mu\otimes\gamma_{c})(E)=\int_{X}\int_{X}\mathbf{1}_{E}(x,y)\,\gamma_{c}(dy)\,\mu(dx)=\int_{X}\int_{X}\mathbf{1}_{E}(x,y)\,f(x)\,\gamma_{c}(dy)\,\gamma_{c}(dx)=\int_{X\times X}\mathbf{1}_{E}\,\tilde{f}\,d(\gamma_{c}\otimes\gamma_{c}),$$

so $\tilde{f}$ is a density of $\mu\otimes\gamma_{c}$ with respect to $\gamma_{c}\otimes\gamma_{c}$. Moreover $(\phi\circ\tilde{f})(x,y)=\phi(f(x))$, so $\phi\circ\tilde{f}$ is measurable, Tonelli's theorem gives $\int|\phi\circ\tilde{f}|\,d(\gamma_{c}\otimes\gamma_{c})=\int_{X}|\phi\circ f|\,d\gamma_{c}<\infty$ (the inner integrals being integrals of constants against $\gamma_{c}(X)=1$), and Fubini's theorem gives $\int\phi\circ\tilde{f}\,d(\gamma_{c}\otimes\gamma_{c})=\int_{X}\phi\circ f\,d\gamma_{c}$. Hence, by Relative Entropy of Probability Measures §relative-entropy on the measurable space $(X\times X,\mathcal{B}(X)\otimes\mathcal{B}(X))$, the probability measure $\mu\otimes\gamma_{c}$ has finite relative entropy with respect to $\gamma_{c}\otimes\gamma_{c}$ and $H(\mu\otimes\gamma_{c}\,|\,\gamma_{c}\otimes\gamma_{c})=H(\mu\,|\,\gamma_{c})$. Now apply Relative Entropy on a Measurable Space: the Gibbs Inequality, the Variational Criterion and Formula, the Entropy Inequality, Small Sets and Data Processing §data-processing with $T=M_{t}$, which is measurable from $\mathcal{B}(X)\otimes\mathcal{B}(X)$ to $\mathcal{B}(X)$: its image measures are $(M_{t})_{\#}(\mu\otimes\gamma_{c})=\mu P_{t}$ and $(M_{t})_{\#}(\gamma_{c}\otimes\gamma_{c})=\gamma_{c}P_{t}=\gamma_{c}$, by The Mehler Law Flow of a Probability Measure on a Hilbert Space §law-flow and claim 1. Therefore $\mu P_{t}$ has finite relative entropy with respect to $\gamma_{c}$ and $H(\mu P_{t}\,|\,\gamma_{c})\le H(\mu\otimes\gamma_{c}\,|\,\gamma_{c}\otimes\gamma_{c})=H(\mu\,|\,\gamma_{c})$.