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

Throughout, each result cited is universally quantified over the data appearing in its own statement and is applied to the data named here. Let $\iota$ be the canonical map from $\mathbb{N}$ to $\mathbb{R}$, positive with positive inverse by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. Write $G_{\mu}$ for the gradients of test functions, so that $T_{\mu}=\overline{G_{\mu}}$, and $\lVert\cdot\rVert_{\mu}$ for the norm of $L^{2}(\mu;\mathbb{R}^{d})$.

For $K\in\mathbb{N}$ put $A_{K}=\{x\in\mathbb{R}^{d}:\lVert T(x)\rVert\le\iota(K)\}$, a Borel set because $x\mapsto\lVert T(x)\rVert^{2}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv) and $A_{K}$ is the preimage of a ray under it, by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field.

Step 1 (Choice of the truncation level). Let $\varepsilon$ be a positive real number; it is fixed first.

The sets $A_{K}$ increase with $K$, by claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and their union is $\mathbb{R}^{d}$, since for each $x$ claim 1 of The Archimedean Property of the Real Numbers gives $K$ with $\lVert T(x)\rVert<\iota(K)$. Hence $(\mu(A_{K}))_{K}$ converges to $\mu(\mathbb{R}^{d})=1$ by claim 5 of Basic Properties of a Measure. Writing $B_{K}=\mathbb{R}^{d}\setminus A_{K}$ and $u_{K}=\lVert T\rVert^{2}\mathbf{1}_{B_{K}}$, the functions $u_{K}$ are Borel, the factor $\lVert T\rVert^{2}$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), the indicator of $B_{K}$ by that lemma and the product by claim 3 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps; they satisfy $0\le u_{K}\le\lVert T\rVert^{2}$ and converge pointwise to $0$ by the same two facts, so $\bigl(\int u_{K}\,d\mu\bigr)_{K}$ converges to $0$ by claim 7 of The Lebesgue Integral and Null Sets, $\lVert T\rVert^{2}$ being integrable with respect to $\mu$ by hypothesis.

Choose $K\in\mathbb{N}$ with $\int_{\mathbb{R}^{d}}u_{K}\,d\mu\le\tfrac{\varepsilon^{2}}{16}$ and $0<\mu(A_{K})$, and put $L=\iota(K)$, a positive real number. Since $\mu(D)=1$ and $\mu$ is a probability measure, $\mu(A_{K}\cap D)=\mu(A_{K})>0$ by claims 2 and 3 of Basic Properties of a Measure, so $A_{K}\cap D$ is nonempty; fix $x_{0}$ in it and put $q_{0}=T(x_{0})$, so that $q_{0}\in\partial_{G}\phi(x_{0})$ and $\lVert q_{0}\rVert\le L$.

Step 2 (Truncation and mollification of the potential). The data $G$, $\phi$, $L$, $x_{0}$, $q_{0}$ satisfy the hypotheses of The Lipschitz Truncation of a Convex Function: a Global Lipschitz Convex Minorant Agreeing with It Where the Slope is Small, applied in dimension $n=d$; let $\phi^{L}:\mathbb{R}^{d}\to\mathbb{R}$ be the Lipschitz truncation of $\phi$ at level $L$. By The Lipschitz Truncation of a Convex Function: a Global Lipschitz Convex Minorant Agreeing with It Where the Slope is Small (i) it is convex on $\mathbb{R}^{d}$ and Lipschitz with constant $L$, and by The Lipschitz Truncation of a Convex Function: a Global Lipschitz Convex Minorant Agreeing with It Where the Slope is Small (iii), applied at each $x\in A_{K}\cap D$, where $\partial_{G}\phi(x)=\{T(x)\}$ with $\lVert T(x)\rVert\le L$,

$$\partial_{\mathbb{R}^{d}}\phi^{L}(x)=\{T(x)\}\qquad\text{for every }x\in A_{K}\cap D. \tag{$*$}$$

Fix a mollifier kernel $\rho$ of radius $1$ on $\mathbb{R}^{d}$, which exists by Existence of Mollifier Kernels of Every Radius, and for $m\in\mathbb{N}$ let $\phi^{L}_{m}$ be the convolution of $\phi^{L}$ with the rescaled kernel of parameter $\iota(m)^{-1}$, as in Mollification of a Lipschitz Convex Function: Smooth Convex Approximations with Bounded Gradients Converging Where the Subgradient is Unique. By Mollification of a Lipschitz Convex Function: Smooth Convex Approximations with Bounded Gradients Converging Where the Subgradient is Unique (i) each $\phi^{L}_{m}$ is smooth on $\mathbb{R}^{d}$, hence of class $C^{1}$, and satisfies $\lVert D\phi^{L}_{m}(x)\rVert\le L$ for every $x$. The sequence $(\iota(m)^{-1})_{m}$ converges to $0$: given a positive real $\varepsilon'$, claim 3 of The Archimedean Property of the Real Numbers provides $m_{0}\in\mathbb{N}$ with $\iota(m_{0})^{-1}<\varepsilon'$, and every $m\ge m_{0}$ satisfies $\iota(m_{0})\le\iota(m)$ by claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, hence $\iota(m)^{-1}\le\iota(m_{0})^{-1}<\varepsilon'$ on multiplying by $\iota(m_{0})^{-1}\iota(m)^{-1}$, nonnegative by claims 7 and 5 of Elementary Order Arithmetic in an Ordered Field, using claim 5 of Elementary Arithmetic in an Ordered Field and claim 2 of Elementary Order Arithmetic in an Ordered Field. So Mollification of a Lipschitz Convex Function: Smooth Convex Approximations with Bounded Gradients Converging Where the Subgradient is Unique (ii) together with $(*)$ gives, for every $x\in A_{K}\cap D$,

$$\bigl(D\phi^{L}_{m}(x)\bigr)_{m\in\mathbb{N}}\ \text{converges to}\ T(x). \tag{$**$}$$

Step 3 (The mollified gradients are tangent). Fix $m$ and put $g_{m}=D\phi^{L}_{m}$. By claim 4 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, $|\partial_{i}\phi^{L}_{m}(x)|\le\lVert D\phi^{L}_{m}(x)\rVert\le L$ for every $x\in\mathbb{R}^{d}$ and every $i\in[d]$. Hence Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields (i), applied to the function $\phi^{L}_{m}$ of class $C^{1}$ with the bound $M=L$, gives that $g_{m}$ is Borel with $\int\lVert g_{m}\rVert^{2}\,d\mu\le dL^{2}<\infty$ and that its class satisfies

$$g_{m}\in T_{\mu}.$$

Step 4 (Convergence in mean square). The functions $v_{m}=\lVert g_{m}-T\rVert^{2}\mathbf{1}_{A_{K}\cap D}$ are Borel, the factor $\lVert g_{m}-T\rVert^{2}$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), the indicator of the Borel set $A_{K}\cap D$ being measurable by that lemma, and the product by claim 3 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps; converge pointwise to $0$ by $(**)$ together with Continuity of the Projections and of the Distance Function on a Product Metric Space, and satisfy $0\le v_{m}\le2L^{2}+2\lVert T\rVert^{2}$ by the inequality $\lVert a-b\rVert^{2}\le2\lVert a\rVert^{2}+2\lVert b\rVert^{2}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv); the dominating function is integrable, by claim 6 of Borel Measurability and Bounded Integration on a Metric Space for the constant and by hypothesis for $\lVert T\rVert^{2}$. So $\bigl(\int v_{m}\,d\mu\bigr)_{m}$ converges to $0$ by claim 7 of The Lebesgue Integral and Null Sets.

Off $A_{K}\cap D$ the measure is carried by $B_{K}$: indeed $\mathbb{R}^{d}\setminus(A_{K}\cap D)$ is the union of $B_{K}$ and $\mathbb{R}^{d}\setminus D$, the latter of measure $0$. On $B_{K}$ one has $L<\lVert T\rVert$, hence $L^{2}\le\lVert T\rVert^{2}$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, so $2L^{2}+2\lVert T\rVert^{2}\le4\lVert T\rVert^{2}$ there. Therefore, by claims 1 and 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and claim 3 of The Lebesgue Integral and Null Sets,

$$\lVert g_{m}-T\rVert_{\mu}^{2}\le\int_{\mathbb{R}^{d}}v_{m}\,d\mu+4\int_{\mathbb{R}^{d}}u_{K}\,d\mu\le\int_{\mathbb{R}^{d}}v_{m}\,d\mu+\tfrac{\varepsilon^{2}}{4}.$$

Choosing $m$ with $\int v_{m}\,d\mu\le\tfrac{\varepsilon^{2}}{4}$ gives $\lVert g_{m}-T\rVert_{\mu}^{2}\le\tfrac{\varepsilon^{2}}{2}\le\varepsilon^{2}$, hence $\lVert g_{m}-T\rVert_{\mu}\le\varepsilon$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. The order of choice is $\varepsilon$, then $K$ and $L$, then $m$.

Step 5 (Conclusion). Let $\varepsilon$ be a positive real number. By Step 4, applied with the positive real number $\tfrac{\varepsilon}{2}$, there is $g\in T_{\mu}$ with $\lVert T-g\rVert_{\mu}\le\tfrac{\varepsilon}{2}$. Since $T_{\mu}=\overline{G_{\mu}}$, claim 3 of Characterization of the Closure in a Metric Space by Open Balls, applied in the metric space $L^{2}(\mu;\mathbb{R}^{d})$ to the point $g$ and the set $G_{\mu}$ with the positive real number $\tfrac{\varepsilon}{2}$, gives $a\in G_{\mu}$ with $\lVert g-a\rVert_{\mu}<\tfrac{\varepsilon}{2}$. By claim 1 of The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity,

$$\lVert T-a\rVert_{\mu}\le\lVert T-g\rVert_{\mu}+\lVert g-a\rVert_{\mu}<\varepsilon .$$

As $\varepsilon$ was arbitrary, claim 3 of Characterization of the Closure in a Metric Space by Open Balls gives $T\in\overline{G_{\mu}}=T_{\mu}$, which is claim 1.