The Gradient of a Convex Continuously Differentiable Function Belongs to the Tangent Space When It Is Square-Integrable

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, with the space $L^{2}(\mu;\mathbb{R}^{d})$ of square-integrable vector fields and the tangent space $T_{\mu}$. Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{d}$ in the sense of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, $\mathbb{R}^{d}$ being open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, with gradient $Df(x)$ at $x$, and convex on $\mathbb{R}^{d}$, the set $\mathbb{R}^{d}$ being convex since every convex combination of two of its points is again one of its points; let $\nabla f:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the gradient map $x\mapsto Df(x)$.

1. (Tangency)¶ The map $\nabla f$ is Borel. If $\int_{\mathbb{R}^{d}}\lVert\nabla f\rVert^{2}\,d\mu<\infty$, then the class of $\nabla f$ in $L^{2}(\mu;\mathbb{R}^{d})$, again written $\nabla f$, belongs to $T_{\mu}$.