The Weighted Fisher Information Relative to a Diagonal Gaussian Measure on Euclidean Space

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $c$ be a variance vector and $\gamma_{c}$ the diagonal Gaussian measure with variances $c$, and let $a=(a_{1},\dots,a_{d})\in\mathbb{R}^{d}$ with $a_{i}>0$ for every $i\in[d]$. Finite Fisher information relative to $\gamma_{c}$ and the relative score $\zeta^{c}_{\nu}$, an element of the space $L^{2}(\nu;\mathbb{R}^{d})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, are those of that definition.

(Weighted relative Fisher information)¶ Let $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ have finite Fisher information relative to $\gamma_{c}$, and let $\zeta=(\zeta_{1},\dots,\zeta_{d}):\mathbb{R}^{d}\to\mathbb{R}^{d}$ be a representative of $\zeta^{c}_{\nu}$, a Borel map with $\int_{\mathbb{R}^{d}}\lVert\zeta\rVert^{2}\,d\nu<\infty$ by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu. The Fisher information of $\nu$ relative to $\gamma_{c}$ with weights $a$ is

$$\mathcal{I}_{a}(\nu\,|\,\gamma_{c})=\int_{\mathbb{R}^{d}}\sum_{i=1}^{d}a_{i}\,\zeta_{i}(y)^{2}\,\nu(dy).$$

The integrand is Borel, its components $\zeta_{i}$ being Borel by claims 2 and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets and its sums and products being Borel by Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; it is nonnegative and at most $\bar{a}\lVert\zeta(y)\rVert^{2}=\bar{a}\sum_{i=1}^{d}\zeta_{i}(y)^{2}$ at every $y$, where $\bar{a}$ is the largest of $a_{1},\dots,a_{d}$, which exists by Greatest Element of a Finite Family in a Totally Ordered Set, so $\mathcal{I}_{a}(\nu\,|\,\gamma_{c})$ is a nonnegative real number by Linearity and Monotonicity of the Lebesgue Integral §nonnegative; and it does not depend on the representative $\zeta$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison.