Proof of 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

Each result cited is universally quantified over the data in its own statement. We write $e_{1},\dots,e_{n}$ for the standard basis vectors of $\mathbb{R}^{n}$, so that $\lVert e_{i}\rVert=1$ and the $i$th coordinate of $z$ is $z\cdot e_{i}$, and for $M\in\mathcal{S}(n)$ we use $e_{i}\cdot(Me_{j})=M_{ij}$, which is claim 4 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum with $w=e_{i}$, $z=e_{j}$.

Claim 1. By Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §convexity, $f$ is semiconvex on $U$ with constant $0$, since $f(x)+\tfrac{0}{2}\lVert x\rVert^{2}=f(x)$. Hence, by Alexandrov's Theorem for Semiconvex Functions on an Open Convex Set §ae, the set $N_{0}$ of points of $U$ at which $f$ is not twice differentiable is null: there is $N\in\mathcal{B}(\mathbb{R}^{n})$ with $N_{0}\subseteq N$ and $\lambda_{n}(N)=0$. Put $A=U\setminus N=U\cap(\mathbb{R}^{n}\setminus N)$, which belongs to $\mathcal{B}(\mathbb{R}^{n})$ since $U$ and $N$ do (Euclidean Space and Lebesgue Measure: Standing Notation §borel) and a $\sigma$-algebra is closed under complements and countable, hence finite, intersections (Sigma-Algebra and Measurable Space). Every point of $A$ lies in $U\setminus N_{0}$, so $f$ is twice differentiable there, and $U\setminus A=U\cap N\subseteq N$ gives $\lambda_{n}(U\setminus A)\le\lambda_{n}(N)=0$ by monotonicity (claim 2 of Basic Properties of a Measure).

Two preliminary facts. (a) $f$ is continuous at every point of $U$: every point of the open set $U$ is an interior point of $U$, so A Convex Function is Continuous near an Interior Point gives a closed ball around it, contained in $U$, on which $f$ is continuous, and continuity at the centre follows since the open ball of the same radius is a neighbourhood of the centre. (b) Let $W\subseteq\mathbb{R}^{n}$ be open, let $F:W\to\mathbb{R}$ be continuous on $W$, and let $\tilde F:\mathbb{R}^{n}\to\mathbb{R}$ equal $F$ on $W$ and $0$ off $W$. Then $\tilde F$ is Borel: for real $a$, the set $\{x\in W:F(x)>a\}$ is open relative to $W$, hence open in $\mathbb{R}^{n}$ as $W$ is open, and $\{\tilde F>a\}$ is this set, united with the closed set $\mathbb{R}^{n}\setminus W$ when $a<0$; both are in $\mathcal{B}(\mathbb{R}^{n})$ by Euclidean Space and Lebesgue Measure: Standing Notation §borel, so $\tilde F$ is Borel by claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line.

Claim 2. Fix $y\in A$ and write $p=Df(y)$, $B=D^{2}f(y)$. Let $\eta>0$ and let $\delta>0$ be as in the definition of twice differentiability of $f$ at $y$ for $\eta$. For $a\in\mathbb{R}^{n}$ with $\lVert a\rVert\le2$ and $m\in\mathbb{N}$ with $2m^{-1}<\delta$ (which holds for all large $m$ by The Archimedean Property of the Real Numbers) we have $y+m^{-1}a\in U$ and

Gradient. Fix $i\in[n]$. For $m\in\mathbb{N}$ let $W_{m}=\{x\in U:x+m^{-1}e_{i}\in U\}$, open as the intersection of $U$ with a translate of $U$, and $F_{m}(x)=m\bigl(f(x+m^{-1}e_{i})-f(x)\bigr)$ for $x\in W_{m}$, continuous on $W_{m}$ by (a). Let $u_{m}=\mathbf{1}_{A}\tilde F_{m}$, Borel by (b) and claims 1 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. For $y\notin A$, $u_{m}(y)=0=g_{i}(y)$. For $y\in A$ and $m$ large, (1) with $a=e_{i}$ gives $y\in W_{m}$ and $|u_{m}(y)-p\cdot e_{i}|\le m^{-1}\bigl(\tfrac12|B_{ii}|+\eta\bigr)$, so $u_{m}(y)\to g_{i}(y)$. By claim 5 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, $g_{i}$ is Borel.

Hessian entries. Fix $i,j\in[n]$. For $m\in\mathbb{N}$ let $W'_{m}$ be the set of $x\in U$ with $x+m^{-1}e_{i}$, $x+m^{-1}e_{j}$ and $x+m^{-1}(e_{i}+e_{j})$ in $U$, again open, and

continuous on $W'_{m}$ by (a); let $v_{m}=\mathbf{1}_{A}\tilde Q_{m}$, Borel as before. For $y\notin A$, $v_{m}(y)=0=H_{ij}(y)$. For $y\in A$ and $m$ large, apply (1) with $a=e_{i}+e_{j}$, $a=e_{i}$ and $a=e_{j}$, all of norm at most $2$ by the triangle inequality (claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$). The terms $f(y)$ cancel; the first-order terms cancel because $p\cdot(e_{i}+e_{j})=p\cdot e_{i}+p\cdot e_{j}$; and the second-order terms give $\tfrac12\bigl((e_{i}+e_{j})\cdot(B(e_{i}+e_{j}))-e_{i}\cdot(Be_{i})-e_{j}\cdot(Be_{j})\bigr)=\tfrac12(B_{ij}+B_{ji})=B_{ij}$, by linearity of the matrix-vector product (claim 3 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum), bilinearity and symmetry of the dot product (Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$), and symmetry of $B$. Hence $|v_{m}(y)-B_{ij}|\le m^{2}\cdot\eta m^{-2}(4+1+1)=6\eta$ for all large $m$. As $\eta$ was arbitrary, $v_{m}(y)\to H_{ij}(y)$, and $H_{ij}$ is Borel by claim 5 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions.

Determinant and trace. By Determinant of a Real Square Matrix and Trace of a Real Square Matrix,

for $y\in A$, and also for $y\notin A$, where every $H_{kl}(y)$ is $0$ and so both right-hand sides vanish (each product has at least one factor, $1\le n$). Thus $J$ and $\Delta$ are finite linear combinations of finite products of Borel functions, hence Borel by claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions.

Claim 3. For $y\notin A$, $J(y)=\Delta(y)=0$. For $y\in A$, Alexandrov's Theorem for Semiconvex Functions on an Open Convex Set §hessian-bound with constant $0$ gives $0_{n}=(-0)I_{n}\preceq D^{2}f(y)$, that is, $0\le z\cdot(D^{2}f(y)z)$ for every $z$, so $D^{2}f(y)$ is positive semidefinite. Then $0\le\det D^{2}f(y)=J(y)$ by Hadamard's Inequality for a Positive Semidefinite Matrix, and each diagonal entry $(D^{2}f(y))_{ii}=e_{i}\cdot(D^{2}f(y)e_{i})$ is nonnegative, so $0\le\Delta(y)$ as a finite sum of nonnegative numbers (claim 5 of Properties of Finite Sums).