Proof of The Hubbard-Stratonovich Transform Couples the Ising Measure and the Field Law: the Density of the Field Law, the Disintegration of the Joint Law, and Entropy Contraction under Both Kernels

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

Conventions. Write $c=c^{J,\eta}$, a variance sequence by The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §variances, and $\gamma=\gamma_{c}$. Through The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §gaussian and A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §background, the notation of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation is in force with the Hilbert space $X$ and its orthonormal basis $(e_{j})_{j\in\mathbb{N}}$, with coordinates $x_{j}=\langle x,e_{j}\rangle_{H^{-m}}$; so The Cameron-Martin Space of a Diagonal Gaussian Measure on a Hilbert Space, The Paley-Wiener Functional of a Vector Relative to a Variance Sequence and The Cameron-Martin Translation Formula for a Diagonal Gaussian Measure on a Hilbert Space apply to $c$. Let $d_{M}$ and $\chi_{M}$ be the Galerkin head dimension and the indicator of the active indices of The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §head-dimension, read with $M$ in place of $N$; thus $\kappa(j)\notin\Gamma_{M}$ for $d_{M}<j$, and for $j\in[d_{M}]$, $\chi_{M}(j)=1$ if $\kappa(j)\in\Gamma_{M}$ and $\chi_{M}(j)=0$ otherwise. For $j\in\mathbb{N}$, $a_{j}^{1/2}$ is the positive square root of $a_{j}$ and $a_{j}^{-1/2}$ its inverse (A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §heads), so $a_{j}^{1/2}a_{j}^{-1}=a_{j}^{-1/2}$. Put $p_{J}(\sigma)=Z_{J}^{-1}\exp(-\mathcal{H}_{J}(\sigma))$ for $\sigma\in\mathcal{C}_{M}$; by The Ising Measure on the Lattice Torus with a Reflection-Symmetric Periodic Pair Interaction §ising, $\mathbb{P}_{J}=\lambda_{p_{J}}$, $\mathbb{P}_{J}(\{\sigma\})=p_{J}(\sigma)>0$ and $\sum_{\sigma\in\mathcal{C}_{M}}p_{J}(\sigma)=1$. For $x\in X$ and $\sigma\in\mathcal{C}_{M}$, $K^{\downarrow}(x,\{\sigma\})=\lambda_{\omega_{x}}(\{\sigma\})=\omega_{x}(\sigma)$ by Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §measure, $\omega_{x}(\sigma)>0$ and $\sum_{\sigma'\in\mathcal{C}_{M}}\omega_{x}(\sigma')=1$, as recorded in The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §field-to-spin; hence $\omega_{x}(\sigma)\le1$ by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §term. Every map $\mathcal{C}_{M}\to\mathbb{R}$ is measurable with respect to $2^{\mathcal{C}_{M}}$ by Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §integral. Five facts are used repeatedly.

(B) A bounded function measurable on a measurable space is integrable with respect to every probability measure on it, by Relative Entropy on a Measurable Space: the Gibbs Inequality, the Variational Criterion and Formula, the Entropy Inequality, Small Sets and Data Processing §functional.

(L) $\mathcal{C}_{M}$ is a nonempty finite set (The Ising Measure on the Lattice Torus with a Reflection-Symmetric Periodic Pair Interaction §configurations), so it has $m$ elements for some $m\in\mathbb{N}$ and there is a bijection $\theta:[m]\to\mathcal{C}_{M}$ (Finite Set, Number of Elements of a Set); by claims 2 and 1 of Properties of a Sum over a Finite Index Set, $\sum_{\sigma\in\mathcal{C}_{M}}t(\sigma)=\sum_{i=1}^{m}t(\theta(i))$ for every map $t:\mathcal{C}_{M}\to\mathbb{R}$. Hence, by Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §integrable and Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear: if $f_{\sigma}$ ($\sigma\in\mathcal{C}_{M}$) are integrable with respect to a measure and $b_{\sigma}$ are real numbers, then $x\mapsto\sum_{\sigma\in\mathcal{C}_{M}}b_{\sigma}f_{\sigma}(x)$ is integrable with integral $\sum_{\sigma\in\mathcal{C}_{M}}b_{\sigma}\int f_{\sigma}$.

(M) If $0<s\le t$ then $\log s\le\log t$: otherwise $\log t<\log s$, and since $\exp$ is strictly increasing (claim 4 of Basic Properties of the Exponential Function), $t=\exp(\log t)<\exp(\log s)=s$ by The Natural Logarithm.

(T) For all real $u$ and $a$, $\exp(u)=\exp(a)\exp(u-a)\ge\exp(a)\,(1+u-a)$, by claim 1 of Basic Properties of the Exponential Function, The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §exp and $\exp(a)>0$ (claim 2 of Basic Properties of the Exponential Function).

(I) Let $(Y,\mathcal{G},\mu)$ be a measure space and $f:Y\to\mathbb{R}$ measurable with $0\le f(y)$ for every $y\in Y$. Then $f^{+}=f$ and $f^{-}=0$ in the notation of Integrable Function and the Lebesgue Integral, and $\int_{Y}0\,d\mu=0\cdot\int_{Y}0\,d\mu=0$ by Linearity and Monotonicity of the Lebesgue Integral §nonnegative with $c=0$. So, by Integrable Function and the Lebesgue Integral, $f$ is integrable exactly when its integral as a nonnegative measurable function, in the sense of Lebesgue Integral of a Nonnegative Measurable Function, is finite, and then the two integrals of $f$ are equal.

Step 1 (the Cameron--Martin data of $h_{\sigma}$). Fix $\sigma\in\mathcal{C}_{M}$; Steps 1 and 2 concern this $\sigma$. For $k\in\mathbb{Z}^{n}$ put $s(k)=L^{-n}\sum_{z\in\mathbb{L}_{M}}\sigma(z)\,\psi_{k}(z)$, so that $s(k)=\hat{\sigma}(k)$ for $k\in\Gamma_{M}$ by The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field §coefficients.

(1a) For every $y\in X$,

$$\sum_{j=1}^{d_{M}}\chi_{M}(j)\,a_{j}^{-1/2}\,y_{j}\,s(\kappa(j))=L^{-n/2}\sum_{z\in\mathbb{L}_{M}}\sigma(z)\,\varphi_{y}(z).$$

Indeed, substitute the definition of $s$, read the sum over $j$ as a sum over $[d_{M}]$ (claim 1 of Properties of a Sum over a Finite Index Set), move constant factors through the sums (claim 4 of Properties of a Sum over a Finite Index Set) and interchange the two sums (claim 5 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set): the left side equals $L^{-n}\sum_{z\in\mathbb{L}_{M}}\sigma(z)\,u(z)$ with $u(z)=\sum_{j=1}^{d_{M}}\chi_{M}(j)\,a_{j}^{-1/2}\,y_{j}\,\psi_{\kappa(j)}(z)$, and $u(z)=L^{n/2}\varphi_{y}(z)$ by The White-Noise Embedding of Lattice Fields: Linearity, Inversion by the Site Field, the Site Sum as the White-Noise Norm, and Coordinates §coordinates; finally $L^{-n}L^{n/2}=L^{-n/2}$, since $L^{n/2}L^{n/2}=L^{n}$.

(1b) Let $j\in\mathbb{N}$. Since $h_{\sigma}\in X$ (The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §spin-to-field), $(h_{\sigma})_{j}=a_{j}^{1/2}\,h_{\sigma}(\kappa(j))$ by The White-Noise Embedding of Lattice Fields: Linearity, Inversion by the Site Field, the Site Sum as the White-Noise Norm, and Coordinates §coordinates. If $\kappa(j)\notin\Gamma_{M}$, in particular if $d_{M}<j$, then $h_{\sigma}(\kappa(j))=0$ and so $(h_{\sigma})_{j}=0$. If $\kappa(j)\in\Gamma_{M}$, then $j\le d_{M}$, $\chi_{M}(j)=1$, $h_{\sigma}(\kappa(j))=\mathfrak{q}_{\kappa(j)}L^{n/2}s(\kappa(j))$ and $c_{j}=a_{j}\,\mathfrak{q}_{\kappa(j)}$ (The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §variances). Hence, for every $j\in[d_{M}]$, both sides vanishing when $\chi_{M}(j)=0$,

$$\frac{(h_{\sigma})_{j}}{c_{j}}=L^{n/2}\,\chi_{M}(j)\,a_{j}^{-1/2}\,s(\kappa(j)).\qquad(1.1)$$

(1c) Let $y\in X$ and let $N\in\mathbb{N}$ with $d_{M}\le N$. The terms $(h_{\sigma})_{j}y_{j}/c_{j}$ with $d_{M}<j\le N$ vanish by (1b), so by A Finite Sum of Vectors with Vanishing Tail (in the real vector space $\mathbb{R}$), (1.1), claim 3 of Properties of Finite Sums and (1a),

$$\sum_{j=1}^{N}\frac{(h_{\sigma})_{j}\,y_{j}}{c_{j}}=\sum_{j=1}^{d_{M}}\frac{(h_{\sigma})_{j}\,y_{j}}{c_{j}}=L^{n/2}\sum_{j=1}^{d_{M}}\chi_{M}(j)\,a_{j}^{-1/2}\,y_{j}\,s(\kappa(j))=\sum_{z\in\mathbb{L}_{M}}\sigma(z)\,\varphi_{y}(z).$$

So the sequence indexed by $N$ on the left is constant from $d_{M}$ on and converges to $\sum_{z\in\mathbb{L}_{M}}\sigma(z)\varphi_{y}(z)$ (Limit of a Sequence of Real Numbers). For $y=h_{\sigma}$ these are the partial sums of the series $\sum_{j=1}^{\infty}(h_{\sigma})_{j}^{2}/c_{j}$ (Series of Real Numbers §partial-sums); so $h_{\sigma}\in H_{c}$ by The Cameron-Martin Space of a Diagonal Gaussian Measure on a Hilbert Space §space, and $|h_{\sigma}|_{c}^{2}=\sum_{z\in\mathbb{L}_{M}}\sigma(z)\,\varphi_{h_{\sigma}}(z)$ by The Cameron-Martin Space of a Diagonal Gaussian Measure on a Hilbert Space §square and Series of Real Numbers §convergent. For $y=x\in X$ they are the Paley--Wiener partial sums $\ell_{h_{\sigma},N}(x)$ of The Paley-Wiener Functional of a Vector Relative to a Variance Sequence §partial-sums, which therefore converge; by The Paley-Wiener Functional of a Vector Relative to a Variance Sequence §functional and claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences,

$$\ell_{h_{\sigma}}(x)=\sum_{z\in\mathbb{L}_{M}}\sigma(z)\,\varphi_{x}(z)\qquad(x\in X).\qquad(1.2)$$

(1d) $|h_{\sigma}|_{c}^{2}=-2\mathcal{H}_{J}(\sigma)+\eta L^{n}$. By The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution §site-field and claim 4 of Properties of a Sum over a Finite Index Set, $\varphi_{h_{\sigma}}(z)=L^{-n/2}\sum_{k\in\Gamma_{M}}\mathfrak{q}_{k}L^{n/2}\hat{\sigma}(k)\psi_{k}(z)=\sum_{k\in\Gamma_{M}}\mathfrak{q}_{k}\hat{\sigma}(k)\psi_{k}(z)$. Interchanging sums (claim 5 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set), using $\sum_{z\in\mathbb{L}_{M}}\sigma(z)\psi_{k}(z)=L^{n}\hat{\sigma}(k)$ (The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field §coefficients), $\mathfrak{q}_{k}=\hat{J}(k)+\eta$, and claims 3 and 4 of Properties of a Sum over a Finite Index Set,

$$\sum_{z\in\mathbb{L}_{M}}\sigma(z)\,\varphi_{h_{\sigma}}(z)=\sum_{k\in\Gamma_{M}}\mathfrak{q}_{k}\hat{\sigma}(k)\,L^{n}\hat{\sigma}(k)=L^{n}\sum_{k\in\Gamma_{M}}\hat{J}(k)\hat{\sigma}(k)\hat{\sigma}(k)+\eta L^{n}\sum_{k\in\Gamma_{M}}\hat{\sigma}(k)\hat{\sigma}(k).$$

By The Interaction Matrix of a Reflection-Symmetric Periodic Pair Interaction is Symmetric and Diagonal in the Trigonometric Basis, with Eigenvalues Its Symbol §energy with $f=g=\sigma$ and The Ising Measure on the Lattice Torus with a Reflection-Symmetric Periodic Pair Interaction §hamiltonian, the first term is $\sum_{y\in\mathbb{L}_{M}}\sum_{z\in\mathbb{L}_{M}}J(y,z)\sigma(y)\sigma(z)=-2\mathcal{H}_{J}(\sigma)$. By Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum §parseval with $f=g=\sigma$, $\sigma(z)\sigma(z)=1$ (as $\sigma(z)\in\{-1,1\}$) and Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum §count-sites, $\sum_{k\in\Gamma_{M}}\hat{\sigma}(k)\hat{\sigma}(k)=L^{-n}\sum_{z\in\mathbb{L}_{M}}1=L^{-n}L^{n}=1$. With (1c) this gives (1d).

(1e) By (1c), $h_{\sigma}\in H_{c}$, and the translation $\tau_{h_{\sigma}}$ of The Cameron-Martin Translation Formula for a Diagonal Gaussian Measure on a Hilbert Space is $T_{h_{\sigma}}$. By (1.2) and (1d) its density is

$$\rho_{h_{\sigma}}(x)=\exp\Bigl(\sum_{z\in\mathbb{L}_{M}}\sigma(z)\,\varphi_{x}(z)+\mathcal{H}_{J}(\sigma)-\frac{\eta L^{n}}{2}\Bigr)\qquad(x\in X),$$

and by The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §spin-to-field and The Cameron-Martin Translation Formula for a Diagonal Gaussian Measure on a Hilbert Space §translation, $K^{\uparrow}_{J,\eta}(\sigma,B)=\int_{X}\mathbf{1}_{B}\,\rho_{h_{\sigma}}\,d\gamma$ for every $B\in\mathcal{B}(X)$. Moreover $\mathbf{1}_{B}\rho_{h_{\sigma}}$ is integrable with respect to $\gamma$: $\mathbf{1}_{B}$ is Borel (claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), $x\mapsto\mathbf{1}_{B}(x+h_{\sigma})$ is the indicator of $T_{h_{\sigma}}^{-1}(B)$, a Borel set as $T_{h_{\sigma}}$ is Borel (The Cameron-Martin Translation Formula for a Diagonal Gaussian Measure on a Hilbert Space §density), hence bounded and Borel and integrable by (B), and The Cameron-Martin Translation Formula for a Diagonal Gaussian Measure on a Hilbert Space §integrable applies with $f=\mathbf{1}_{B}$.

Step 2 (the weights). With $\sigma$ as in Step 1, let $w_{\sigma}:X\to\mathbb{R}$ be the map

$$w_{\sigma}(x)=\frac{1}{Z_{J}}\exp\Bigl(-\frac{\eta L^{n}}{2}+\sum_{z\in\mathbb{L}_{M}}\sigma(z)\,\varphi_{x}(z)\Bigr).$$

Then, for every $x\in X$,

$$p_{J}(\sigma)\,\rho_{h_{\sigma}}(x)=w_{\sigma}(x)=\varrho^{\mathrm{HS}}_{J,\eta}(x)\,\omega_{x}(\sigma).\qquad(2.1)$$

The first equality follows from (1e) and $\exp(-\mathcal{H}_{J}(\sigma))\exp(v)=\exp(v-\mathcal{H}_{J}(\sigma))$ (claim 1 of Basic Properties of the Exponential Function). For the second, by claim 1 of Basic Properties of the Exponential Function, $\varrho^{\mathrm{HS}}_{J,\eta}(x)\,\omega_{x}(\sigma)$ is $Z_{J}^{-1}$ times $\exp$ of $-\eta L^{n}/2+\sum_{z\in\mathbb{L}_{M}}\operatorname{lch}(\varphi_{x}(z))+\sum_{z\in\mathbb{L}_{M}}\bigl(\sigma(z)\varphi_{x}(z)-\operatorname{lch}(\varphi_{x}(z))\bigr)$, and the last sum equals $\sum_{z\in\mathbb{L}_{M}}\sigma(z)\varphi_{x}(z)-\sum_{z\in\mathbb{L}_{M}}\operatorname{lch}(\varphi_{x}(z))$ by claims 3 and 4 of Properties of a Sum over a Finite Index Set. The map $w_{\sigma}$ is continuous with respect to the distance of $X$: each $x\mapsto\varphi_{x}(z)$ is continuous by The White-Noise Embedding of Lattice Fields: Linearity, Inversion by the Site Field, the Site Sum as the White-Noise Norm, and Coordinates §site-continuous, so $x\mapsto\sum_{z\in\mathbb{L}_{M}}\sigma(z)\varphi_{x}(z)$ is continuous by Finite Linear Combinations of Continuous Real-Valued Maps on a Metric Space are Continuous §continuous, and adding the constant $-\eta L^{n}/2$ keeps $x\mapsto-\eta L^{n}/2+\sum_{z\in\mathbb{L}_{M}}\sigma(z)\varphi_{x}(z)$ continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §constants and Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set; $\exp$ is continuous by Differentiability at an Interior Point Implies Continuity There and claim 3 of Basic Properties of the Exponential Function; compositions are continuous by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map, and the multiple by $Z_{J}^{-1}$ by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set.

Since $\sigma$ was arbitrary, (2.1) holds for every $\sigma\in\mathcal{C}_{M}$, and by claim 4 of Properties of a Sum over a Finite Index Set and $\sum_{\sigma\in\mathcal{C}_{M}}\omega_{x}(\sigma)=1$,

$$\sum_{\sigma\in\mathcal{C}_{M}}w_{\sigma}(x)=\varrho^{\mathrm{HS}}_{J,\eta}(x)\qquad(x\in X).\qquad(2.2)$$

Step 3 (the basic identity). For every $A\subseteq\mathcal{C}_{M}$ and every $B\in\mathcal{B}(X)$, the function $\mathbf{1}_{B}K^{\downarrow}(\cdot,A)\,\varrho^{\mathrm{HS}}_{J,\eta}$ is integrable with respect to $\gamma$ and

$$\sum_{\sigma\in\mathcal{C}_{M}}\mathbf{1}_{A}(\sigma)\,p_{J}(\sigma)\,K^{\uparrow}_{J,\eta}(\sigma,B)=\int_{X}\mathbf{1}_{B}(x)\,K^{\downarrow}(x,A)\,\varrho^{\mathrm{HS}}_{J,\eta}(x)\,\gamma(dx).\qquad(3.1)$$

Indeed, by (1e), (2.1) and Linearity and Monotonicity of the Lebesgue Integral §integrable, for each $\sigma$ the function $\mathbf{1}_{B}w_{\sigma}=p_{J}(\sigma)\mathbf{1}_{B}\rho_{h_{\sigma}}$ is integrable with $\int_{X}\mathbf{1}_{B}w_{\sigma}\,d\gamma=p_{J}(\sigma)K^{\uparrow}_{J,\eta}(\sigma,B)$. By (L) with $b_{\sigma}=\mathbf{1}_{A}(\sigma)$, the left side of (3.1) is the integral of the integrable function $x\mapsto\sum_{\sigma\in\mathcal{C}_{M}}\mathbf{1}_{A}(\sigma)\mathbf{1}_{B}(x)w_{\sigma}(x)$. By (2.1) and claim 4 of Properties of a Sum over a Finite Index Set this function equals $\mathbf{1}_{B}(x)\varrho^{\mathrm{HS}}_{J,\eta}(x)\sum_{\sigma\in\mathcal{C}_{M}}\mathbf{1}_{A}(\sigma)\omega_{x}(\sigma)$, and $\sum_{\sigma\in\mathcal{C}_{M}}\mathbf{1}_{A}(\sigma)\omega_{x}(\sigma)=\int_{\mathcal{C}_{M}}\mathbf{1}_{A}\,d\lambda_{\omega_{x}}=\lambda_{\omega_{x}}(A)=K^{\downarrow}(x,A)$ by Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §integral, The Integral of an Indicator Function is the Measure of the Set (whose nonnegative integral equals the integral of Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §integral by (I), $\mathbf{1}_{A}$ being nonnegative) and The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §field-to-spin.

Claim 1 (density). Positivity: $Z_{J}>0$ by The Ising Measure on the Lattice Torus with a Reflection-Symmetric Periodic Pair Interaction §partition and $\exp$ is positive (claim 2 of Basic Properties of the Exponential Function). Continuity: by (2.2), $\varrho^{\mathrm{HS}}_{J,\eta}$ is a finite sum of the maps $w_{\sigma}$, continuous by Step 2, hence continuous by Finite Linear Combinations of Continuous Real-Valued Maps on a Metric Space are Continuous §continuous (with all coefficients $1$); in particular it is Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space. Let $B\in\mathcal{B}(X)$. Take $A=\mathcal{C}_{M}$ in (3.1): $\mathbf{1}_{\mathcal{C}_{M}}(\sigma)=1$ and $K^{\downarrow}(x,\mathcal{C}_{M})=1$, $K^{\downarrow}(x,\cdot)$ being a probability measure. By The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §field-law, The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §laws with $\lambda=\mathbb{P}_{J}$, and $\mathbb{P}_{J}(\{\sigma\})=p_{J}(\sigma)$,

$$\gamma^{\mathrm{HS}}_{J,\eta}(B)=\sum_{\sigma\in\mathcal{C}_{M}}p_{J}(\sigma)\,K^{\uparrow}_{J,\eta}(\sigma,B)=\int_{X}\mathbf{1}_{B}\,\varrho^{\mathrm{HS}}_{J,\eta}\,d\gamma,$$

the integrand being integrable. With $B=X$, $\varrho^{\mathrm{HS}}_{J,\eta}=\mathbf{1}_{X}\varrho^{\mathrm{HS}}_{J,\eta}$ is integrable and $\int_{X}\varrho^{\mathrm{HS}}_{J,\eta}\,d\gamma=\gamma^{\mathrm{HS}}_{J,\eta}(X)=1$, as $\gamma^{\mathrm{HS}}_{J,\eta}\in\mathcal{P}(X)$.

Claim 2 (disintegration). Let $A\subseteq\mathcal{C}_{M}$ and $B\in\mathcal{B}(X)$. By The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §joint, Integration Against a Probability Kernel: Measurable Sections, the Composite Measure on the Product and the Iterated Integral §composite and Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §integral (with $\mathbb{P}_{J}=\lambda_{p_{J}}$), then (3.1),

$$\Xi_{J,\eta}(A\times B)=\int_{\mathcal{C}_{M}}\mathbf{1}_{A}(\sigma)\,K^{\uparrow}_{J,\eta}(\sigma,B)\,\mathbb{P}_{J}(d\sigma)=\sum_{\sigma\in\mathcal{C}_{M}}\mathbf{1}_{A}(\sigma)\,p_{J}(\sigma)\,K^{\uparrow}_{J,\eta}(\sigma,B)=\int_{X}\mathbf{1}_{B}\,K^{\downarrow}(\cdot,A)\,\varrho^{\mathrm{HS}}_{J,\eta}\,d\gamma.\qquad(3.2)$$

The map $\varrho^{\mathrm{HS}}_{J,\eta}:X\to[0,\infty)$ is Borel (Claim 1), so by claim 3 of Image Measures, Measures with Densities, and Change of Variables the measure $\nu_{\varrho}$ with density $\varrho^{\mathrm{HS}}_{J,\eta}$ with respect to $\gamma$ is defined, and by Claim 1 it takes the same value as $\gamma^{\mathrm{HS}}_{J,\eta}$ on every $B\in\mathcal{B}(X)$: the nonnegative integral $\nu_{\varrho}(B)=\int_{X}\mathbf{1}_{B}\,\varrho^{\mathrm{HS}}_{J,\eta}\,d\gamma$ equals the real integral of the nonnegative integrable function $\mathbf{1}_{B}\varrho^{\mathrm{HS}}_{J,\eta}$ in Claim 1 by (I), so $\nu_{\varrho}=\gamma^{\mathrm{HS}}_{J,\eta}$. The function $f=\mathbf{1}_{B}K^{\downarrow}(\cdot,A)$ is Borel by claims 1 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, $x\mapsto K^{\downarrow}(x,A)$ being Borel by The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §field-to-spin, and $f\varrho^{\mathrm{HS}}_{J,\eta}$ is integrable with respect to $\gamma$ by Step 3. Hence, by claim 3 of Image Measures, Measures with Densities, and Change of Variables, $f$ is integrable with respect to $\gamma^{\mathrm{HS}}_{J,\eta}$ and $\int_{X}f\varrho^{\mathrm{HS}}_{J,\eta}\,d\gamma=\int_{X}f\,d\gamma^{\mathrm{HS}}_{J,\eta}$, which with (3.2) is Claim 2.

Claim 3 (marginals). Let $A\subseteq\mathcal{C}_{M}$. By The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §laws with $\nu=\gamma^{\mathrm{HS}}_{J,\eta}$, Claim 2 with $B=X$ (where $\mathbf{1}_{X}=1$), the middle expression of (3.2), $K^{\uparrow}_{J,\eta}(\sigma,X)=1$ (The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §spin-to-field), Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §integral, The Integral of an Indicator Function is the Measure of the Set and (I) for the nonnegative $\mathbf{1}_{A}$,

$$(\gamma^{\mathrm{HS}}_{J,\eta}K^{\downarrow})(A)=\int_{X}K^{\downarrow}(x,A)\,\gamma^{\mathrm{HS}}_{J,\eta}(dx)=\Xi_{J,\eta}(A\times X)=\sum_{\sigma\in\mathcal{C}_{M}}\mathbf{1}_{A}(\sigma)\,p_{J}(\sigma)=\int_{\mathcal{C}_{M}}\mathbf{1}_{A}\,d\mathbb{P}_{J}=\mathbb{P}_{J}(A).$$

Both sides being maps on $2^{\mathcal{C}_{M}}$, $\gamma^{\mathrm{HS}}_{J,\eta}K^{\downarrow}=\mathbb{P}_{J}$.

Claim 4 (contraction under $K^{\downarrow}$). Fix $\nu$ as in the claim; $\nu K^{\downarrow}$ is a probability measure on $(\mathcal{C}_{M},2^{\mathcal{C}_{M}})$ by The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §laws. Then let $F:\mathcal{C}_{M}\to\mathbb{R}$ be an arbitrary bounded map, measurable by the conventions, with a bound $b$ (Bounded Real-Valued Function on a Set); the function $G$ below is built from $F$.

For $x\in X$ put $S(x)=\sum_{\sigma\in\mathcal{C}_{M}}\omega_{x}(\sigma)\exp(F(\sigma))$. As $-b\le F(\sigma)\le b$ and $\exp$ is increasing (claim 4 of Basic Properties of the Exponential Function), $\exp(-b)\,\omega_{x}(\sigma)\le\omega_{x}(\sigma)\exp(F(\sigma))\le\exp(b)\,\omega_{x}(\sigma)$; summing, by Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §comparison, claim 4 of Properties of a Sum over a Finite Index Set and $\sum_{\sigma\in\mathcal{C}_{M}}\omega_{x}(\sigma)=1$, $\exp(-b)\le S(x)\le\exp(b)$. So $S(x)>0$, and $G(x)=\log S(x)$ satisfies $-b\le G(x)\le b$ by (M) and The Natural Logarithm. The map $S$ is a finite sum of constant multiples of the continuous maps $x\mapsto\omega_{x}(\sigma)$ (The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §field-to-spin), hence continuous by Finite Linear Combinations of Continuous Real-Valued Maps on a Metric Space are Continuous §continuous with the coefficients $\exp(F(\sigma))$; $\log$ is continuous on $(0,\infty)$ by Differentiability at an Interior Point Implies Continuity There and The Natural Logarithm; so $G$ is continuous by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map, hence Borel (claim 3 of Borel Measurability and Bounded Integration on a Metric Space). Thus $G$ is bounded measurable on $(X,\mathcal{B}(X))$.

Put $a(x)=\sum_{\sigma\in\mathcal{C}_{M}}\omega_{x}(\sigma)F(\sigma)$. By (T) with $u=F(\sigma)$ and $a=a(x)$, multiplied by $\omega_{x}(\sigma)\ge0$ and summed (Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §comparison, claims 3 and 4 of Properties of a Sum over a Finite Index Set),

$$S(x)\ge\exp(a(x))\Bigl(\sum_{\sigma\in\mathcal{C}_{M}}\omega_{x}(\sigma)+\sum_{\sigma\in\mathcal{C}_{M}}\omega_{x}(\sigma)F(\sigma)-a(x)\sum_{\sigma\in\mathcal{C}_{M}}\omega_{x}(\sigma)\Bigr)=\exp(a(x)),$$

so $G(x)\ge a(x)$ by (M) and The Natural Logarithm.

By the converse part of Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §measure, $\nu K^{\downarrow}=\lambda_{q}$ with $q(\sigma)=(\nu K^{\downarrow})(\{\sigma\})=\int_{X}\omega_{x}(\sigma)\,\nu(dx)$ (The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §laws), the map $x\mapsto\omega_{x}(\sigma)$ being continuous by The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §field-to-spin, hence Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space, with values in $(0,1]$, hence integrable by (B). By Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §integral, (L), $G\ge a$, (B) for $G$ and Linearity and Monotonicity of the Lebesgue Integral §integrable,

$$\int_{\mathcal{C}_{M}}F\,d(\nu K^{\downarrow})=\sum_{\sigma\in\mathcal{C}_{M}}F(\sigma)\int_{X}\omega_{x}(\sigma)\,\nu(dx)=\int_{X}a\,d\nu\le\int_{X}G\,d\nu.$$

By Claim 3 and The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §laws, $p_{J}(\sigma)=\mathbb{P}_{J}(\{\sigma\})=(\gamma^{\mathrm{HS}}_{J,\eta}K^{\downarrow})(\{\sigma\})=\int_{X}\omega_{x}(\sigma)\,\gamma^{\mathrm{HS}}_{J,\eta}(dx)$; so by Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §integral, (L) and $\exp(\log S(x))=S(x)$ (The Natural Logarithm),

$$\int_{\mathcal{C}_{M}}\exp\circ F\,d\mathbb{P}_{J}=\sum_{\sigma\in\mathcal{C}_{M}}\exp(F(\sigma))\,p_{J}(\sigma)=\int_{X}S\,d\gamma^{\mathrm{HS}}_{J,\eta}=\int_{X}\exp\circ G\,d\gamma^{\mathrm{HS}}_{J,\eta}.$$

Subtracting the logarithms of these equal positive numbers, and then using Relative Entropy on a Measurable Space: the Gibbs Inequality, the Variational Criterion and Formula, the Entropy Inequality, Small Sets and Data Processing §gibbs on $(X,\mathcal{B}(X))$ for $\nu$, $\gamma^{\mathrm{HS}}_{J,\eta}$ and the bounded measurable $G$,

$$\Lambda^{\mathbb{P}_{J}}_{F}(\nu K^{\downarrow})\le\Lambda^{\gamma^{\mathrm{HS}}_{J,\eta}}_{G}(\nu)\le H(\nu\,|\,\gamma^{\mathrm{HS}}_{J,\eta}).$$

As $F$ was an arbitrary bounded measurable map on $(\mathcal{C}_{M},2^{\mathcal{C}_{M}})$, Relative Entropy on a Measurable Space: the Gibbs Inequality, the Variational Criterion and Formula, the Entropy Inequality, Small Sets and Data Processing §criterion with the constant $H(\nu\,|\,\gamma^{\mathrm{HS}}_{J,\eta})$ shows that $\nu K^{\downarrow}$ has finite relative entropy with respect to $\mathbb{P}_{J}$ and $H(\nu K^{\downarrow}\,|\,\mathbb{P}_{J})\le H(\nu\,|\,\gamma^{\mathrm{HS}}_{J,\eta})$.

Claim 5 (contraction under $K^{\uparrow}_{J,\eta}$). (5a) Integration against $\mu K^{\uparrow}_{J,\eta}$. Let $\mu$ be a probability measure on $(\mathcal{C}_{M},2^{\mathcal{C}_{M}})$ and $F:X\to\mathbb{R}$ bounded and Borel. Then $F$ is integrable with respect to each $K^{\uparrow}_{J,\eta}(\sigma,\cdot)$ by (B), and

$$\int_{X}F\,d(\mu K^{\uparrow}_{J,\eta})=\sum_{\sigma\in\mathcal{C}_{M}}\mu(\{\sigma\})\int_{X}F\,dK^{\uparrow}_{J,\eta}(\sigma,\cdot).\qquad(5.1)$$

Indeed, let $\pi_{2}:\mathcal{C}_{M}\times X\to X$, $\pi_{2}(\sigma,y)=y$. For $E\in\mathcal{B}(X)$, $\pi_{2}^{-1}(E)=\mathcal{C}_{M}\times E$ is a measurable rectangle, so it belongs to $2^{\mathcal{C}_{M}}\otimes\mathcal{B}(X)$ (Product Sigma-Algebra); thus $\pi_{2}$ is measurable, and by The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §laws, $(\mu K^{\uparrow}_{J,\eta})(E)=(\mu\otimes K^{\uparrow}_{J,\eta})(\pi_{2}^{-1}(E))$, so $\mu K^{\uparrow}_{J,\eta}$ is the image measure of $\mu\otimes K^{\uparrow}_{J,\eta}$ under $\pi_{2}$ (claim 1 of Image Measures, Measures with Densities, and Change of Variables). As $\mu K^{\uparrow}_{J,\eta}\in\mathcal{P}(X)$ (The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §laws), $F$ is integrable with respect to it by (B), and claim 2 of Image Measures, Measures with Densities, and Change of Variables gives $\int_{X}F\,d(\mu K^{\uparrow}_{J,\eta})=\int F\circ\pi_{2}\,d(\mu\otimes K^{\uparrow}_{J,\eta})$. The function $F\circ\pi_{2}$ is measurable with respect to $2^{\mathcal{C}_{M}}\otimes\mathcal{B}(X)$ by claim 4 of Borel Measurability and Bounded Integration on a Metric Space and bounded by any bound of $F$; so by Integration Against a Probability Kernel: Measurable Sections, the Composite Measure on the Product and the Iterated Integral §bounded the last integral equals $\int_{\mathcal{C}_{M}}g\,d\mu$ with $g(\sigma)=\int_{X}F\,dK^{\uparrow}_{J,\eta}(\sigma,\cdot)$. Finally $\mu=\lambda_{p_{\mu}}$ with $p_{\mu}(\sigma)=\mu(\{\sigma\})$ by the converse part of Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §measure, and Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §integral gives $\int_{\mathcal{C}_{M}}g\,d\mu=\sum_{\sigma\in\mathcal{C}_{M}}\mu(\{\sigma\})\,g(\sigma)$.

(5b) Fix $\lambda$ as in the claim. By the converse part of Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §measure, $\lambda=\lambda_{p}$ with $p(\sigma)=\lambda(\{\sigma\})\ge0$, and $\sum_{\sigma\in\mathcal{C}_{M}}p(\sigma)=1$ by the same claim, $\lambda_{p}$ being a probability measure; also $p_{J}(\sigma)>0$ and $\sum_{\sigma\in\mathcal{C}_{M}}p_{J}(\sigma)=1$. By Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §entropy, $\lambda$ has finite relative entropy with respect to $\lambda_{p_{J}}=\mathbb{P}_{J}$.

(5c) With $\lambda$ fixed, let $F:X\to\mathbb{R}$ be an arbitrary bounded Borel function, with a bound $b$; the function $G$ below is built from $F$. For $\sigma\in\mathcal{C}_{M}$ put $E(\sigma)=\int_{X}\exp\circ F\,dK^{\uparrow}_{J,\eta}(\sigma,\cdot)$, where $\exp\circ F$ is bounded measurable by Relative Entropy on a Measurable Space: the Gibbs Inequality, the Variational Criterion and Formula, the Entropy Inequality, Small Sets and Data Processing §functional. Since $\exp(-b)\le\exp(F(y))\le\exp(b)$ for $y\in X$ (claim 4 of Basic Properties of the Exponential Function) and constants $t$ have integral $t$ against the probability measure $K^{\uparrow}_{J,\eta}(\sigma,\cdot)$ (claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space), monotonicity (Linearity and Monotonicity of the Lebesgue Integral §integrable) gives $\exp(-b)\le E(\sigma)\le\exp(b)$. So $G(\sigma)=\log E(\sigma)$ is defined and $-b\le G(\sigma)\le b$ by (M) and $\log(\exp(\pm b))=\pm b$ (The Natural Logarithm); $G$ is bounded, and measurable by the conventions.

Put $a(\sigma)=\int_{X}F\,dK^{\uparrow}_{J,\eta}(\sigma,\cdot)$. By (T), $\exp(F(y))\ge\exp(a(\sigma))\bigl(1+F(y)-a(\sigma)\bigr)$ for every $y\in X$; the right side is integrable in $y$ with integral $\exp(a(\sigma))\bigl(1+a(\sigma)-a(\sigma)\bigr)=\exp(a(\sigma))$, by Linearity and Monotonicity of the Lebesgue Integral §integrable and claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space. By monotonicity, $E(\sigma)\ge\exp(a(\sigma))$, so $G(\sigma)\ge a(\sigma)$ by (M) and The Natural Logarithm.

By (5.1) with $\mu=\lambda$, Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits §comparison (as $\lambda(\{\sigma\})\ge0$) and Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §integral with $\lambda=\lambda_{p}$,

$$\int_{X}F\,d(\lambda K^{\uparrow}_{J,\eta})=\sum_{\sigma\in\mathcal{C}_{M}}\lambda(\{\sigma\})\,a(\sigma)\le\sum_{\sigma\in\mathcal{C}_{M}}\lambda(\{\sigma\})\,G(\sigma)=\int_{\mathcal{C}_{M}}G\,d\lambda .$$

By The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §field-law, (5.1) with $\mu=\mathbb{P}_{J}$ and $\exp\circ F$ in place of $F$, $\exp(\log E(\sigma))=E(\sigma)$ and Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §integral with $\mathbb{P}_{J}=\lambda_{p_{J}}$,

$$\int_{X}\exp\circ F\,d\gamma^{\mathrm{HS}}_{J,\eta}=\sum_{\sigma\in\mathcal{C}_{M}}p_{J}(\sigma)\,E(\sigma)=\sum_{\sigma\in\mathcal{C}_{M}}p_{J}(\sigma)\exp(G(\sigma))=\int_{\mathcal{C}_{M}}\exp\circ G\,d\mathbb{P}_{J}.$$

Subtracting the logarithms of these equal positive numbers, and then using Relative Entropy on a Measurable Space: the Gibbs Inequality, the Variational Criterion and Formula, the Entropy Inequality, Small Sets and Data Processing §gibbs on $(\mathcal{C}_{M},2^{\mathcal{C}_{M}})$ for $\lambda$, $\mathbb{P}_{J}$ (legitimate by (5b)) and the bounded measurable $G$,

$$\Lambda^{\gamma^{\mathrm{HS}}_{J,\eta}}_{F}(\lambda K^{\uparrow}_{J,\eta})\le\Lambda^{\mathbb{P}_{J}}_{G}(\lambda)\le H(\lambda\,|\,\mathbb{P}_{J}).$$

As $F$ was an arbitrary bounded measurable function on $(X,\mathcal{B}(X))$ and $\lambda K^{\uparrow}_{J,\eta}\in\mathcal{P}(X)$, Relative Entropy on a Measurable Space: the Gibbs Inequality, the Variational Criterion and Formula, the Entropy Inequality, Small Sets and Data Processing §criterion with the constant $H(\lambda\,|\,\mathbb{P}_{J})$ shows that $\lambda K^{\uparrow}_{J,\eta}$ has finite relative entropy with respect to $\gamma^{\mathrm{HS}}_{J,\eta}$ and $H(\lambda K^{\uparrow}_{J,\eta}\,|\,\gamma^{\mathrm{HS}}_{J,\eta})\le H(\lambda\,|\,\mathbb{P}_{J})$. Together with (5b) this is Claim 5.