The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, let $V$ be an admissible cylindrical potential, let $b$ be a constant as in Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §below, and let $\beta\in\mathbb{R}$ be positive; $\exp$ is the exponential function, with the properties listed in Basic Properties of the Exponential Function.

(The weight)¶ Write $w_{V,\beta}=\exp(-V/\beta):X\to\mathbb{R}$. It is Borel and satisfies $0<w_{V,\beta}(x)\le\exp(b/\beta)$ for every $x\in X$. Indeed, by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity, $V$ is continuous on $X$ and $-b\le V(x)$ for every $x\in X$; $\exp$ is a smooth map on $\mathbb{R}$ by claim 3 of Basic Properties of the Exponential Function, and so continuous, whence $w_{V,\beta}$ is continuous, hence Borel; $w_{V,\beta}(x)>0$ by claim 2 of that theorem; and $-V(x)/\beta\le b/\beta$, $\beta$ being positive, so $w_{V,\beta}(x)\le\exp(b/\beta)$ since $\exp$ is increasing by claim 4 of that theorem.

(The normaliser)¶ The normaliser of $V$ at temperature $\beta$ is

$$Z_{V,\beta}=\int_{X}w_{V,\beta}\,d\gamma_{c},$$

a positive real number. Indeed, by the clause on the weight above and the monotonicity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative, applied with $f=w_{V,\beta}$ and $g$ the constant function $\exp(b/\beta)$, $\int_{X}w_{V,\beta}\,d\gamma_{c}\le\exp(b/\beta)\,\gamma_{c}(X)=\exp(b/\beta)<\infty$, $\gamma_{c}$ being a probability measure; so the nonnegative Borel function $w_{V,\beta}$ is integrable with respect to $\gamma_{c}$ and $Z_{V,\beta}$ is a nonnegative real number. It is not $0$: otherwise $w_{V,\beta}(x)=0$ for $\gamma_{c}$-almost every $x$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §vanishing, whereas $w_{V,\beta}>0$ everywhere on $X$ by the clause on the weight above and $\gamma_{c}(X)=1$.

(The Gibbs measure)¶ The Gibbs measure of $V$ at temperature $\beta$ relative to $\gamma_{c}$ is the function $\gamma^{V}_{\beta}$ on $\mathcal{B}(X)$,

$$\gamma^{V}_{\beta}(A)=\frac{1}{Z_{V,\beta}}\int_{X}\mathbf{1}_{A}\,w_{V,\beta}\,d\gamma_{c}\qquad(A\in\mathcal{B}(X)),$$

with $Z_{V,\beta}$ the normaliser of the preceding clause. Here $\mathbf{1}_{A}w_{V,\beta}$ is Borel, as a product of Borel functions, and $0\le\mathbf{1}_{A}w_{V,\beta}\le w_{V,\beta}$. It is a Borel probability measure on $X$: each value is a real number in $[0,1]$, since $\int_{X}\mathbf{1}_{A}w_{V,\beta}\,d\gamma_{c}\le Z_{V,\beta}$ by the monotonicity of Linearity and Monotonicity of the Lebesgue Integral §nonnegative; $\gamma^{V}_{\beta}(X)=Z_{V,\beta}^{-1}Z_{V,\beta}=1$ and $\gamma^{V}_{\beta}(\emptyset)=0$; and for pairwise disjoint $A_{1},A_{2},\dots\in\mathcal{B}(X)$ with union $A$ the partial sums $\sum_{i=1}^{m}\mathbf{1}_{A_{i}}w_{V,\beta}=\mathbf{1}_{A_{1}\cup\dots\cup A_{m}}w_{V,\beta}$ increase pointwise to $\mathbf{1}_{A}w_{V,\beta}$, so that countable additivity follows from Monotone Convergence Theorem and the additivity of the integral of Linearity and Monotonicity of the Lebesgue Integral §nonnegative.