The Gibbs Measure of a Borel Potential Relative to a Diagonal Gaussian Measure

In the settings of The Real Numbers: Standing Notation and Background and A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, let $V:X\to\mathbb{R}$ be a Borel function and let $\beta\in\mathbb{R}$ be positive, such that the function $x\mapsto\exp(-V(x)/\beta)$ on $X$ is integrable with respect to $\gamma_{c}$; here $\exp$ 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: $-V/\beta$ is Borel by Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and $\exp$ is a smooth map on $\mathbb{R}$ by claim 3 of Basic Properties of the Exponential Function and so continuous, so the composition is Borel by claim 4 of Borel Measurability and Bounded Integration on a Metric Space; and $0<w_{V,\beta}(x)$ for every $x\in X$ by claim 2 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, $w_{V,\beta}$ is nonnegative and integrable by hypothesis, so $Z_{V,\beta}$ is a nonnegative real number; and it is not $0$, since 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 $w_{V,\beta}>0$ on $X$ 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 $\mathcal{B}(X)$,

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

It is a Borel probability measure on $X$: for $A\in\mathcal{B}(X)$ the function $\mathbf{1}_{A}w_{V,\beta}$ is Borel with $0\le\mathbf{1}_{A}w_{V,\beta}\le w_{V,\beta}$, so each value lies in $[0,1]$ 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 $\gamma^{V}_{\beta}(\emptyset)=0$ since $\mathbf{1}_{\emptyset}w_{V,\beta}=0$; 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 countable additivity follows from Monotone Convergence Theorem and the additivity of the integral in Linearity and Monotonicity of the Lebesgue Integral §nonnegative.