The Points of Twice Differentiability of a Convex Function: a Borel Set of Full Measure, and Borel Measurability of the Gradient and Hessian on It

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, used with a natural number $n$ satisfying $1\le n$, let $U\subseteq\mathbb{R}^{n}$ be open and convex and let $f:U\to\mathbb{R}$ be convex on $U$. At a point $y\in U$ at which $f$ is twice differentiable, $Df(y)$ denotes the first-order coefficient, which is the gradient of $f$ at $y$ as recorded there, and $D^{2}f(y)\in\mathcal{S}(n)$ the Hessian. For a real $n\times n$ matrix $M$, $\det M$ is its determinant and $\mathrm{tr}\,M$ its trace.

1. (A Borel set of full measure)¶ There is $A\in\mathcal{B}(\mathbb{R}^{n})$ with $A\subseteq U$ and $\lambda_{n}(U\setminus A)=0$ such that $f$ is twice differentiable at every point of $A$; here $U\setminus A$ belongs to $\mathcal{B}(\mathbb{R}^{n})$, $U$ being open.

For the remaining claims let $A\in\mathcal{B}(\mathbb{R}^{n})$ satisfy $A\subseteq U$, suppose that $f$ is twice differentiable at every point of $A$, and for $i,j\in[n]$ let $g_{i},H_{ij},J,\Delta:\mathbb{R}^{n}\to\mathbb{R}$ be the functions whose values at $y\in A$ are, respectively, the $i$th coordinate of $Df(y)$, the entry of $D^{2}f(y)$ in row $i$ and column $j$, $\det D^{2}f(y)$ and $\mathrm{tr}\,D^{2}f(y)$, and whose values at every $y\in\mathbb{R}^{n}\setminus A$ are $0$.

2. (Borel derivatives)¶ The functions $g_{i}$, $H_{ij}$ ($i,j\in[n]$), $J$ and $\Delta$ are Borel.

3. (Nonnegativity)¶ $0\le J(y)$ and $0\le\Delta(y)$ for every $y\in\mathbb{R}^{n}$.