A Square-Integrable Selection of the Subdifferential of a Convex Potential Belongs to the Tangent Space

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, let $L^{2}(\mu;\mathbb{R}^{d})$ be the space of square-integrable vector fields against $\mu$ and let $T_{\mu}$ be the tangent space at $\mu$.

Let $G\subseteq\mathbb{R}^{d}$ be open, hence a member of $\mathcal{B}(\mathbb{R}^{d})$ by Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background (iv), and convex, let $\phi:G\to\mathbb{R}$ be convex on $G$, with subdifferential $\partial_{G}\phi$, and let $D\in\mathcal{B}(\mathbb{R}^{d})$ satisfy $D\subseteq G$ and $\mu(D)=1$, so that $\mu(G)=1$ by claim 2 of Basic Properties of a Measure and in particular $G$ is nonempty. Let $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be Borel with

$$\partial_{G}\phi(x)=\{T(x)\}\qquad\text{for every }x\in D,$$

and suppose that $\int_{\mathbb{R}^{d}}\lVert T\rVert^{2}\,d\mu<\infty$, so that the class of $T$ belongs to $L^{2}(\mu;\mathbb{R}^{d})$ and is again written $T$.

1. (Tangency)(i) $T\in T_{\mu}$.