Integrals of Bounded $C^2$ Cylindrical Functions are Noise Intrinsic Test Functions with the Noise Gradient as Gradient

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $\mathcal{P}^{a}_{\rho}$ be the set of The Measures Noise-Connected to the Reference Measure §space. Let $n\in\mathbb{N}$, let $f\in C^{2}_{b}(\mathbb{R}^{n})$, the set of Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded with $q=n$, and let $\varphi=f\circ p_{n}$, a bounded $C^{2}$ cylindrical function and hence a bounded $C^{1}$ cylindrical function by Noise Gradients of Bounded C^2 Cylindrical Functions Are Dense in the Noise Tangent Space §inclusion. For $\mu\in\mathcal{P}(X)$ the function $\varphi$ is integrable with respect to $\mu$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §square-integrable, and the class of its noise gradient $\nabla_{a}\varphi$ in $L^{2}(\mu;X^{a})$, again written $\nabla_{a}\varphi$, lies in the noise tangent space $T^{a}_{\mu}$ by Noise Gradients of Bounded C^2 Cylindrical Functions Are Dense in the Noise Tangent Space §inclusion. Let $U_{\varphi}:\mathcal{P}^{a}_{\rho}\to\mathbb{R}$ be the function

$$U_{\varphi}(\mu)=\int_{X}\varphi\,d\mu .$$

Noise intrinsic test functions and the gradient along noise couplings $\nabla U_{\varphi}(\mu)$ are those of those definitions.

1. (Test function)¶ For every subset $Q\subseteq\mathcal{P}^{a}_{\rho}$, $U_{\varphi}$ is a noise intrinsic test function on $Q$.

2. (Gradient)¶ For every $\mu\in\mathcal{P}^{a}_{\rho}$, $U_{\varphi}$ is differentiable along noise couplings at $\mu$ and $\nabla U_{\varphi}(\mu)=\nabla_{a}\varphi$ in $L^{2}(\mu;X^{a})$.