The Intrinsic Fisher Information Relative to the Gibbs Measure of a Gibbs-Sobolev Potential

In the settings of The Real Numbers: Standing Notation and Background and Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, let $\beta,q\in\mathbb{R}$ satisfy $0<\beta$ and $1<q$, let $V$ be a Gibbs-Sobolev potential at temperature $\beta$ with exponent $q$, with Gibbs measure $\gamma^{V}_{\beta}$, Gibbs Sobolev space $\mathbb{D}^{1,2}_{a}(\gamma^{V}_{\beta})$ and Gibbs Dirichlet form $\mathcal{E}^{V}_{a}$, and with $L^{2}(\gamma^{V}_{\beta})$ as in The Gibbs Sobolev Space of the Noise Gradient. For a real number $s\ge0$, $\sqrt{s}$ is its nonnegative square root.

(Intrinsic Fisher information)¶ A measure $\mu\in\mathcal{P}(X)$ has finite intrinsic Fisher information relative to $\gamma^{V}_{\beta}$ with weights $a$ if there is a Borel function $f:X\to[0,\infty)$ with $\mu(A)=\int_{A}f\,d\gamma^{V}_{\beta}$ for every $A\in\mathcal{B}(X)$ (a density of $\mu$ relative to $\gamma^{V}_{\beta}$) such that the class of $\sqrt{f}$ lies in $\mathbb{D}^{1,2}_{a}(\gamma^{V}_{\beta})$. Here $\sqrt{f}=(x\mapsto\sqrt{f(x)})$ is Borel, since for real $t$ the set of the $x$ with $\sqrt{f(x)}>t$ is $X$ if $t<0$ and is the set of the $x$ with $f(x)>t^{2}$ if $0\le t$, by Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; and $(\sqrt{f})^{2}=f$ pointwise by Existence and Uniqueness of the Nonnegative Square Root, so $\int_{X}(\sqrt{f})^{2}\,d\gamma^{V}_{\beta}=\mu(X)=1$ (the defining identity with $A=X$) and its class lies in $L^{2}(\gamma^{V}_{\beta})$. The intrinsic Fisher information of $\mu$ relative to $\gamma^{V}_{\beta}$ with weights $a$ is then the nonnegative real number

$$\mathcal{I}^{V}_{a}(\mu)=4\,\mathcal{E}^{V}_{a}\bigl(\sqrt{f},\sqrt{f}\bigr).$$

It does not depend on the choice of $f$: two densities of $\mu$ relative to $\gamma^{V}_{\beta}$ agree $\gamma^{V}_{\beta}$-almost everywhere by The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities §uniqueness, so their square roots have the same class.