Proof of The Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation with a Linear Running Cost: Its Viscosity Solution Is the Integral of the Pointwise Solution, Which It Determines

Each result cited below is universally quantified over the data in its own statement. Elementary facts about real numbers, finite sums, squares and absolute values are used through The Real Numbers: Standing Notation and Background §background without further mention.

Conventions. Put $M=\lambda_{0}^{-1}C$; since $0\le|\tilde{g}(x)|\le C$ and $\lambda_{0}^{-1}$ is positive, $0\le M$. Integrals against probability measures on $\mathbb{R}^{d}$ are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures; every bounded Borel function is integrable against each of them by that clause. Linearity, monotonicity and $|\int f\,d\mu|\le\int|f|\,d\mu$ are those of Linearity and Monotonicity of the Lebesgue Integral §integrable, and for a constant $b$ and a probability measure $\mu$ we have $\int b\,d\mu=b\,\mu(\mathbb{R}^{d})=b$ by that clause applied to $b\mathbf{1}_{\mathbb{R}^{d}}$ and The Integral of an Indicator Function is the Measure of the Set. The dimension satisfies $1\le d$ by The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §data, and $\mathbb{R}^{d}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous and nonempty since it contains $0_{\mathbb{R}^{d}}$.

Step 1 (The potential). Let $Q_{c}\in\mathcal{S}(d)$ be the diagonal matrix with diagonal entries $a/c_{1},\dots,a/c_{d}$. Then $x\cdot(Q_{c}x)=\sum_{i=1}^{d}(a/c_{i})x_{i}^{2}=a|x|_{c}^{2}$ by The Diagonal Gaussian Density on Euclidean Space and Its Notation §scaling, so $V_{c}(x)=\tfrac12x\cdot(Q_{c}x)$, and by Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic (linear coefficient $0$, constant term $0$), $DV_{c}(x)=Q_{c}x$, the vector with components $(a/c_{i})x_{i}$, and $D^{2}V_{c}(x)=Q_{c}$, whose trace is $\sum_{i=1}^{d}a/c_{i}$ by Trace of a Real Square Matrix. Since every $a/c_{i}$ is positive, $0\le V_{c}(x)$; and since $c_{i}\le c_{\max}$ for every $i$ (The Diagonal Gaussian Density on Euclidean Space and Its Notation §variances),

$$V_{c}(x)=\tfrac{a}{2}\sum_{i=1}^{d}\frac{x_{i}^{2}}{c_{i}}\ \ge\ \frac{a}{2c_{\max}}\lVert x\rVert^{2}\qquad(x\in\mathbb{R}^{d}).$$

For $x,y\in\mathbb{R}^{d}$,

$$\bigl(DV_{c}(x)-DV_{c}(y)\bigr)\cdot(x-y)=\sum_{i=1}^{d}\frac{a}{c_{i}}(x_{i}-y_{i})^{2}\ \ge\ 0, \tag{1}$$

so $DV_{c}$ is monotone.

Step 2 ($V_{c}$ is a penalty satisfying the two conditions). Penalty on an Open Subset of Euclidean Space §regularity holds because $V_{c}$ is of class $C^{2}$ on $\mathbb{R}^{d}$. Let $t\in\mathbb{R}$ and $K_{t}=\{x\in\mathbb{R}^{d}:V_{c}(x)\le t\}$. By claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, $V_{c}$ is continuous on $\mathbb{R}^{d}$, hence lower semicontinuous there by the final assertion of Semicontinuity Under Negation and Characterization of Continuity, so $K_{t}$ is closed by claim 3 of Semicontinuity via Sublevel and Superlevel Sets (space and subset both $\mathbb{R}^{d}$, metric $d_{E}$). For $x\in K_{t}$, Step 1 gives $\lVert x\rVert^{2}\le2c_{\max}a^{-1}|t|=:R_{t}$, so $\lVert x\rVert\le1+R_{t}$ (if $1\le\lVert x\rVert$ then $\lVert x\rVert\le\lVert x\rVert^{2}$); thus $K_{t}$ is bounded, and compact by Heine-Borel Theorem in $\mathbb{R}^n$. So Penalty on an Open Subset of Euclidean Space §sublevel holds and $V_{c}$ is a penalty on $\mathbb{R}^{d}$. By (1), $0\le(DV_{c}(x)-DV_{c}(y))\cdot(x-y)$ for all $x,y$, which is the one-sided condition Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §one-sided with $c_{R}=0$ for every $R$. With noise intensity $2a$ and $C_{1}=a\sum_{i=1}^{d}a/c_{i}$, Step 1 and $0\le\lambda_{0}V_{c}(x)$ give

$$\tfrac{2a}{2}\operatorname{tr}\bigl(D^{2}V_{c}(x)\bigr)=C_{1}\le\lambda_{0}V_{c}(x)+C_{1}\qquad(x\in\mathbb{R}^{d}),$$

which is the dissipation condition Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §dissipation with $\varepsilon=1$, so that the coefficient $1-\varepsilon$ of $\lVert DV_{c}(x)\rVert^{2}$ is $0$, and constant $C_{1}$.

Step 3 (Clause 1). We apply Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality with dimension $d$, open set $\mathbb{R}^{d}$, penalty $V_{c}$ (Step 2), discount $\lambda_{0}$, control cost $\theta$ and noise intensity $2a$ (both nonnegative), running cost $\tilde{g}$, which is continuous by A Uniformly Continuous Map Between Metric Spaces Is Continuous §continuous, and bound $C$. The operator of The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space §operator with these data is given by the same formula as that of The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space §operator, so it is $F$, and in both items viscosity sub- and supersolutions are those of Viscosity Subsolution and Supersolution of a Second-Order Equation. Both conditions hold by Step 2, so by Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §well-posed there is exactly one $w:\mathbb{R}^{d}\to\mathbb{R}$ that is a viscosity subsolution and supersolution of $F$ on $\mathbb{R}^{d}$ with $V_{c}$-subordinate growth from above and below; it is continuous and $|w(x)|\le M$ for every $x$, so $w$ is bounded with bound $M$ (Bounded Real-Valued Function on a Set). Let $w'$ be another bounded function that is a viscosity subsolution and supersolution of $F$ on $\mathbb{R}^{d}$, with bound $B$. For positive $\delta$, $0\le\delta V_{c}(x)$ gives $w'(x)\le B+\delta V_{c}(x)$ and $-B-\delta V_{c}(x)\le w'(x)$, so $w'$ has $V_{c}$-subordinate growth from above and below (Penalty-Subordinate Growth of a Function on an Open Subset of Euclidean Space §above, Penalty-Subordinate Growth of a Function on an Open Subset of Euclidean Space §below, with constant $B$), and $w'=w$ by the uniqueness in Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §well-posed. Finally $w$, being continuous, is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps, and, being bounded and Borel, integrable with respect to every element of $\mathcal{P}(\mathbb{R}^{d})\supseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. This proves clause 1.

Step 4 (A modulus of continuity of $\tilde{g}$). For $s\in[0,\infty)$ let $E_{s}=\{|\tilde{g}(x)-\tilde{g}(y)|:x,y\in\mathbb{R}^{d},\ \lVert x-y\rVert\le s\}$. It contains $0$ (take $x=y$) and is bounded above by $2C$, so it has a least upper bound $\rho(s)$ (Upper Bound and Least Upper Bound), because the real numbers are Dedekind complete. Then $0\le\rho(s)$; if $s\le s'$ then $E_{s}\subseteq E_{s'}$, so $\rho(s)\le\rho(s')$; and $|\tilde{g}(x)-\tilde{g}(y)|\in E_{\lVert x-y\rVert}$ gives $|\tilde{g}(x)-\tilde{g}(y)|\le\rho(\lVert x-y\rVert)$ for all $x,y$. Moreover, for every positive $\alpha$ there is a positive $s_{0}$ with $\rho(s)\le\alpha$ for $0\le s\le s_{0}$: by Uniformly Continuous Map Between Metric Spaces there is a positive $\delta$ with $|\tilde{g}(x)-\tilde{g}(y)|<\alpha$ whenever $d_{E}(x,y)=\lVert x-y\rVert<\delta$; with $s_{0}=\delta/2$, every element of $E_{s}$ ($s\le s_{0}$) is less than $\alpha$, so $\rho(s)\le\alpha$.

Step 5 (Upper bound by sup-convolution). Let $u^{*}:\mathcal{D}\to\mathbb{R}$ be a viscosity solution with $-M\le u^{*}(\mu)\le M$, given by Well-Posedness of the Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §existence; its hypotheses hold since $0<\theta\le1$ and $G$ is bounded by $C$ and uniformly continuous (statement). Fix $\mu\in\mathcal{D}$ and a positive $\beta$. By Step 4 choose a positive $s_{0}$ with $\rho(s)\le\lambda_{0}\beta$ for $0\le s\le s_{0}$, put $\tau=s_{0}^{2}(4M+1)^{-1}>0$, and let $r$ be the nonnegative square root of $4\tau M$. Then $r^{2}=4Ms_{0}^{2}(4M+1)^{-1}\le s_{0}^{2}$, so $0\le r\le s_{0}$ and $b:=\lambda_{0}^{-1}\rho(r)$ satisfies $0\le b\le\beta$.

Apply Part A of Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation on the Whole Space with a Monotone Drift with $n=d$, $P=V_{c}$ (monotone gradient by (1)), discount $\lambda_{0}$, control cost $\theta$, noise intensity $2a$, $g=\tilde{g}$, the modulus $\rho$ of Step 4, the numbers $M,\tau$ (so that its $r$ is ours), and $u=w$ (Step 3, $|w|\le M$); its operator $F$ is ours. This gives $\overline{w}$ with $w\le\overline{w}\le M$ (Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation on the Whole Space with a Monotone Drift §sub-bounds), so $|\overline{w}|\le M$; semiconvex with constant $\tau^{-1}$ (Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation on the Whole Space with a Monotone Drift §sub-semiconvex); Lipschitz with constant $L=\tau^{-1}r$ (Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation on the Whole Space with a Monotone Drift §sub-lipschitz); and a viscosity subsolution of $F_{+}$, the operator with running cost $\tilde{g}+\rho(r)$ (Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation on the Whole Space with a Monotone Drift §subsolution).

Let $w_{1}=\overline{w}-b$. Then $-M-b\le w_{1}\le M$, so $w_{1}$ is bounded; $|w_{1}(x)-w_{1}(y)|=|\overline{w}(x)-\overline{w}(y)|$, so $w_{1}$ is Lipschitz with constant $L$, hence continuous by A Lipschitz Map is Uniformly Continuous and upper semicontinuous by Semicontinuity Under Negation and Characterization of Continuity; and since $(w_{1}+\tfrac{1}{2\tau}\lVert\cdot\rVert^{2})=(\overline{w}+\tfrac{1}{2\tau}\lVert\cdot\rVert^{2})-b$ and $t+(1-t)=1$, the convexity inequality of Convex Real-Valued Function on a Convex Subset of $\mathbb{R}^n$ passes from the latter to the former, so $w_{1}$ is semiconvex with constant $\tau^{-1}$ (Semiconvex Function on a Convex Subset of $\mathbb{R}^n$).

$w_{1}$ is a viscosity subsolution of $F$ on $\mathbb{R}^{d}$. Let $\varphi$ be of class $C^{2}$ on $\mathbb{R}^{d}$ and let $w_{1}-\varphi$ have a local maximum at $x$. The constant function $b$ is of class $C^{2}$ with gradient $0$ and Hessian $0$ by Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic (matrix $0$, linear coefficient $0$, constant term $b$), so $\psi=\varphi+b$ is of class $C^{2}$ by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, and by the sum rule of claim 1 of that lemma, applied to first and then to second partial derivatives, $D\psi(x)=D\varphi(x)$ and $D^{2}\psi(x)=D^{2}\varphi(x)$. As $\overline{w}-\psi=w_{1}-\varphi$ pointwise, $\overline{w}-\psi$ has a local maximum at $x$, so by Viscosity Subsolution and Supersolution of a Second-Order Equation for $F_{+}$, with the formula of The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space §operator, and since $\lambda_{0}b=\rho(r)$ (so that $\lambda_{0}w_{1}(x)=\lambda_{0}\overline{w}(x)-\rho(r)$) and $D\psi=D\varphi$, $D^{2}\psi=D^{2}\varphi$ at $x$,

$$F\bigl(x,w_{1}(x),D\varphi(x),D^{2}\varphi(x)\bigr)=\lambda_{0}\overline{w}(x)-\rho(r)+\tfrac{\theta}{2}\lVert D\psi(x)\rVert^{2}+DV_{c}(x)\cdot D\psi(x)-a\operatorname{tr}(D^{2}\psi(x))-\tilde{g}(x)=F_{+}\bigl(x,\overline{w}(x),D\psi(x),D^{2}\psi(x)\bigr)\le0 .$$

By Integrals of Lipschitz Semiconvex Viscosity Subsolutions and Lipschitz Semiconcave Viscosity Supersolutions of the Ornstein-Uhlenbeck Hamilton-Jacobi Equation Are Viscosity Sub- and Supersolutions of the Lifted Equation §subsolution (with $L$, $K=\tau^{-1}$), $W_{1}(\nu)=\int w_{1}\,d\nu$ is a viscosity subsolution of the lifted equation; $w_{1}$ and $\overline{w}=w_{1}+b$ are integrable against every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by the preamble of that proposition, and $W_{1}(\nu)\le M$ by monotonicity. Since $u^{*}$ is a viscosity supersolution (Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §solution) with $-M\le u^{*}$, Well-Posedness of the Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §comparison gives $W_{1}\le u^{*}$ on $\mathcal{D}$. By monotonicity and linearity,

$$\mathcal{U}(\mu)=\int w\,d\mu\le\int\overline{w}\,d\mu=W_{1}(\mu)+b\le u^{*}(\mu)+\beta .$$

Step 6 (Lower bound by inf-convolution). With $\mu,\beta,\tau,r,b$ as in Step 5, Part B of Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation on the Whole Space with a Monotone Drift with $v=w$ gives $\underline{w}$ with $-M\le\underline{w}\le w$ (Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation on the Whole Space with a Monotone Drift §inf-bounds), $-\underline{w}$ semiconvex with constant $\tau^{-1}$ (Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation on the Whole Space with a Monotone Drift §inf-semiconcave), Lipschitz with constant $L$ (Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation on the Whole Space with a Monotone Drift §inf-lipschitz), and a viscosity supersolution of $F_{-}$, with running cost $\tilde{g}-\rho(r)$ (Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation on the Whole Space with a Monotone Drift §supersolution). Let $w_{2}=\underline{w}+b$. As in Step 5, $w_{2}$ is bounded ($-M\le w_{2}\le M+b$), Lipschitz with constant $L$, lower semicontinuous, and $-w_{2}=-\underline{w}-b$ is semiconvex with constant $\tau^{-1}$. If $\varphi$ is of class $C^{2}$ and $w_{2}-\varphi$ has a local minimum at $x$, then $\psi=\varphi-b$ is of class $C^{2}$ with the same gradient and Hessian (as in Step 5, with the constant $-b$), $\underline{w}-\psi=w_{2}-\varphi$ has a local minimum at $x$, and

$$F\bigl(x,w_{2}(x),D\varphi(x),D^{2}\varphi(x)\bigr)=F_{-}\bigl(x,\underline{w}(x),D\psi(x),D^{2}\psi(x)\bigr)\ge0 ,$$

since $\lambda_{0}w_{2}(x)-\tilde{g}(x)=\lambda_{0}\underline{w}(x)-(\tilde{g}(x)-\rho(r))$. So $w_{2}$ is a viscosity supersolution of $F$ on $\mathbb{R}^{d}$, and by Integrals of Lipschitz Semiconvex Viscosity Subsolutions and Lipschitz Semiconcave Viscosity Supersolutions of the Ornstein-Uhlenbeck Hamilton-Jacobi Equation Are Viscosity Sub- and Supersolutions of the Lifted Equation §supersolution, $W_{2}(\nu)=\int w_{2}\,d\nu$ is a viscosity supersolution of the lifted equation, with $-M\le W_{2}$ by monotonicity of the integral (Linearity and Monotonicity of the Lebesgue Integral §integrable and the conventions), since $-M\le\underline{w}\le\underline{w}+b=w_{2}$. As $u^{*}$ is a viscosity subsolution with $u^{*}\le M$, Well-Posedness of the Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §comparison gives $u^{*}\le W_{2}$ on $\mathcal{D}$, and

$$\mathcal{U}(\mu)=\int w\,d\mu\ge\int\underline{w}\,d\mu=W_{2}(\mu)-b\ge u^{*}(\mu)-\beta .$$

Step 7 (Clause 2). By Steps 5 and 6, $|\mathcal{U}(\mu)-u^{*}(\mu)|\le\beta$ for every positive $\beta$, so $\mathcal{U}(\mu)=u^{*}(\mu)$; as $\mu\in\mathcal{D}$ was arbitrary, $\mathcal{U}=u^{*}$, which is a viscosity solution. If $u:\mathcal{D}\to\mathbb{R}$ is a bounded viscosity solution, then, $u^{*}$ being bounded with bound $M$, Well-Posedness of the Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §uniqueness gives $u=u^{*}=\mathcal{U}$. This proves clause 2.

Step 8 ($\eta_{x,\varepsilon}\in\mathcal{D}$). Let $\varepsilon$ be positive, $e=(\varepsilon,\dots,\varepsilon)$, a variance vector, and $\gamma_{\varepsilon}=\gamma_{e}$, a probability measure with density $\rho_{e}$ with respect to $\lambda_{d}$ (Diagonal Gaussian Measures on Euclidean Space §measure). By claim 3 of Image Measures, Measures with Densities, and Change of Variables and The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms §moments,

$$\int\lVert y\rVert^{2}\,\gamma_{\varepsilon}(dy)=\int\lVert y\rVert^{2}\rho_{e}(y)\,\lambda_{d}(dy)=d\varepsilon<\infty, \tag{2}$$

so $\gamma_{\varepsilon}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ (The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment, The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space). The constant function $1$ is measurable (claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) and nonnegative, and $\int\mathbf{1}_{A}\cdot1\,d\gamma_{\varepsilon}=\gamma_{\varepsilon}(A)$ for every Borel $A$ by The Integral of an Indicator Function is the Measure of the Set, so it is a density of $\gamma_{\varepsilon}$ with respect to itself in the sense of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities. Since $\log1=0$ by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §log with $t=1$, $\phi\circ1$ is the zero function, whose integral is $0$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-integral (null set $\varnothing$); so it is integrable, and $\gamma_{\varepsilon}$ has finite relative entropy with respect to $\gamma_{e}$ (Relative Entropy of Probability Measures §relative-entropy). By Changing the Variances of a Diagonal Gaussian Reference Measure: Equal Domains, the Shift of the Relative Entropy and the Shift of the Relative Score §domains it has finite relative entropy with respect to $\gamma_{c}$, so $\gamma_{\varepsilon}\in\mathcal{D}$ (The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure §pair). The translation $y\mapsto x+y$ is the map $\tau_{x}$ of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §constants, so $\eta_{x,\varepsilon}=(\tau_{x})_{\#}\gamma_{\varepsilon}\in\mathcal{D}$ by The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property §translation.

Step 9 (Clause 3). Fix $x\in\mathbb{R}^{d}$. Each component $y\mapsto x_{k}+y_{k}$ of $\tau_{x}$ is Lipschitz with constant $1$, hence continuous (A Lipschitz Map is Uniformly Continuous) and Borel, so $\tau_{x}$ is Borel and so is $w\circ\tau_{x}$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps). By Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward and the conventions,

$$|\mathcal{U}(\eta_{x,\varepsilon})-w(x)|=\Bigl|\int\bigl(w(x+y)-w(x)\bigr)\gamma_{\varepsilon}(dy)\Bigr|\le\int h\,d\gamma_{\varepsilon},\qquad h(y)=|w(x+y)-w(x)|,$$

$h$ being Borel by claims 1, 2 and 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and bounded by $2M$. Let $\sigma$ be positive. By continuity of $w$ at $x$ (Continuous Map Between Metric Spaces) there is a positive $\delta$ with $h(y)<\sigma$ whenever $d_{E}(x,x+y)=\lVert y\rVert<\delta$. Let $A=\{y:\delta^{2}\le\lVert y\rVert^{2}\}$, a Borel set with $\delta^{2}\gamma_{\varepsilon}(A)\le d\varepsilon$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §markov and (2). If $y\notin A$ then $\lVert y\rVert<\delta$, so $h(y)\le\sigma+2M\mathbf{1}_{A}(y)$ for every $y$, and by monotonicity, linearity and The Integral of an Indicator Function is the Measure of the Set,

$$|\mathcal{U}(\eta_{x,\varepsilon})-w(x)|\le\sigma+2M\gamma_{\varepsilon}(A)\le\sigma+2Md\,\delta^{-2}\varepsilon .$$

Now let $(\varepsilon_{j})$ be positive and converge to $0$. By Limit of a Sequence of Real Numbers there is $N$ with $\varepsilon_{j}<\sigma\delta^{2}(2Md+1)^{-1}$ for $j\ge N$, and then $|\mathcal{U}(\eta_{x,\varepsilon_{j}})-w(x)|<2\sigma$. As $\sigma$ was arbitrary, $(\mathcal{U}(\eta_{x,\varepsilon_{j}}))$ converges to $w(x)$. With Step 8 this proves clause 3.