Proof of Mollifying a Probability Measure on Euclidean Space: Kernel Bounds, the Translation Estimate, the Mollified Density and Its Cost

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here. The field axioms and the rules for adding inequalities, for multiplying or dividing them by positive real numbers, for multiplying them by nonnegative real numbers, and for handling absolute values, from Elementary Order Arithmetic in an Ordered Field, Elementary Arithmetic in an Ordered Field and Properties of the Absolute Value in an Ordered Field, are used without further mention; so are the power rules $c^{1}=c$, $c^{n+1}=c^{n}c$, $(cc')^{n}=c^{n}c'^{n}$ and $1^{n}=1$ of claims 1, 3 and 2 of Properties of Natural Number Powers in a Field, and the fact that $0<c^{n}$ for real $c>0$ (claims 5 and 4 there). Points of $\mathbb{R}^{d}$ are $d$-tuples of reals by Euclidean Points as Tuples of Real Numbers; sums, negatives and differences of points are formed coordinatewise, and the laws of the real vector space $\mathbb{R}^{d}$ are used without mention. For the Euclidean norm we use $\lVert x-y\rVert=d_{E}(x,y)=d_{E}(y,x)$ (claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and Euclidean Distance is a Metric on $\mathbb{R}^n$), the coordinate bound $|x_{i}|\le\lVert x\rVert$ (claim 4 there), homogeneity $\lVert tx\rVert=|t|\,\lVert x\rVert$ (claim 5) and the triangle inequality (claim 6). Throughout, $c_{\varepsilon}=(\varepsilon^{-1})^{d}$, a positive real number, since $0<\varepsilon^{-1}$.

Step 1 (Claim 1).

(1a) Scaling. By Rescaling a Mollifier Kernel, applied with $n=d$, the kernel $\eta$ of radius $\delta=1$ and the given $\varepsilon$, the function $\eta_{\varepsilon}$ is a mollifier kernel of radius $\varepsilon\cdot1=\varepsilon$. Hence, by conditions 1 to 4 of Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$: $\eta$ and $\eta_{\varepsilon}$ are smooth on $\mathbb{R}^{d}$ and nonnegative; $\eta(z)=0$ whenever $1<\lVert z\rVert$ and $\eta_{\varepsilon}(z)=0$ whenever $\varepsilon<\lVert z\rVert$; and both are integrable with respect to $\lambda_{d}$ with integral $1$.

(1b) Continuity. $\mathbb{R}^{d}$ is open in itself by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, so claim 3 of that lemma (with $U=\mathbb{R}^{d}$, $m=1$, in its version for smooth maps) shows that $\eta$ and $\eta_{\varepsilon}$ are continuous relative to $\mathbb{R}^{d}$ at every point, as maps into $(\mathbb{R},d_{\mathbb{R}})$ with the absolute-value metric; this is continuity in the sense of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §extrema, and also continuity into $\mathbb{R}$ with its Euclidean distance, which is $d_{\mathbb{R}}$ by The Euclidean Distance on the Real Line is the Absolute Value Metric. Both functions are therefore Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps.

(1c) Test function. As $\eta(z)=0$ whenever $1<\lVert z\rVert$, claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set (with $R=1$) shows that $\eta$ is compactly supported for the topology of open subsets of $(\mathbb{R}^{d},d_{E})$, which is the topology of Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §space by Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$. Being smooth, $\eta$ is a test function.

(1d) The bound $S$. By claim 1 of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable, which applies by (1b) and (1c), $\eta$ is bounded: there is a real $M\ge0$ with $|\eta(z)|\le M$ for all $z$ (Bounded Real-Valued Function on a Set). Then $S=M+1$ is positive and $\eta(z)\le|\eta(z)|\le M<S$ for every $z$. Now let $S$ be any real number with $\eta(z)\le S$ for all $z$. For $z\in\mathbb{R}^{d}$, (1a) gives $0\le\eta_{\varepsilon}(z)$, and $\eta_{\varepsilon}(z)=c_{\varepsilon}\,\eta(\varepsilon^{-1}z)\le c_{\varepsilon}S$ because $0<c_{\varepsilon}$.

(1e) The bound $D$. By The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient, applied with $q=d$ and $\psi=\eta$, there is a real $K\ge0$ with $\lVert\nabla\eta(z)\rVert\le K$ for every $z$; the $i$th coordinate of $\nabla\eta(z)$ is $\partial_{i}\eta(z)$ by Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §gradient, so $|\partial_{i}\eta(z)|\le\lVert\nabla\eta(z)\rVert\le K$ by the coordinate bound. Thus $D=K$ has the required property.

(1f) Lipschitz bounds. Let $D$ be any real number with $|\partial_{i}\eta(z)|\le D$ for all $z\in\mathbb{R}^{d}$ and $i\in[d]$, and let $z,z'\in\mathbb{R}^{d}$. By Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §gradient, $\eta$ is of class $C^{1}$ on $\mathbb{R}^{d}$, so by clause 1 of C^k Maps on a Euclidean Open Set (read through its scalar convention, clause 3) the partial derivative $\partial_{i}\eta(p)$ exists at every $p\in\mathbb{R}^{d}$ for every $i\in[d]$. Put $w^{0}=z$, and for $j\in[d]$ let $w^{j}\in\mathbb{R}^{d}$ have $i$th coordinate $z'_{i}$ for $i\le j$ and $z_{i}$ for $j<i$; thus $w^{d}=z'$, and for $j\in[d]$ the points $w^{j-1}$ and $w^{j}$ have the same coordinates except the $j$th, which is $z_{j}$ for $w^{j-1}$ and $z'_{j}$ for $w^{j}$.

Claim $(\ast)$: for every $j\in[d]$, $|\eta(w^{j})-\eta(w^{j-1})|\le D\,|z'_{j}-z_{j}|$. If $z_{j}=z'_{j}$ then $w^{j}=w^{j-1}$ and there is nothing to prove. Otherwise let $\alpha<\beta$ be the smaller and the larger of $z_{j},z'_{j}$. For $s\in\mathbb{R}$ let $a[s]\in\mathbb{R}^{d}$ be the point with $j$th coordinate $s$ and all other coordinates those of $w^{j-1}$, so $a[z_{j}]=w^{j-1}$ and $a[z'_{j}]=w^{j}$, and let $g:\mathbb{R}\to\mathbb{R}$, $g(s)=\eta(a[s])$. Fix $s\in\mathbb{R}$ and apply Slice Function and the Partial Derivative with $n=d$, $U=\mathbb{R}^{d}$, $f=\eta$, the point $a[s]$ and $i=j$: the points obtained from $a[s]$ by replacing its $j$th coordinate by $s'$ are the points $a[s']$, which lie in $U=\mathbb{R}^{d}$, so $\rho=1$ is admissible in its claim 1, and the slice function of its claim 2 is the restriction of $g$ to $I=\{s'\in\mathbb{R}:s-1<s'<s+1\}$. Since $\partial_{j}\eta(a[s])$ exists, claim 2 there shows that $g|_{I}$ is differentiable at $s$ with derivative $\partial_{j}\eta(a[s])$. Now $I$ is an interval with $s$ as an interior point by Basic Facts about Intervals of the Real Line and Their Interior Points §open-interval, $\mathbb{R}$ is an interval with $s$ as an interior point by Basic Facts about Intervals of the Real Line and Their Interior Points §whole-line, and $\{s'\in\mathbb{R}:|s'-s|<1\}=I$; so Differentiability at an Interior Point is a Local Property §local shows that $g$ is differentiable at $s$ with $g'(s)=\partial_{j}\eta(a[s])$, whence $|g'(s)|\le D$, and Differentiability at an Interior Point Implies Continuity There shows that $g$ is continuous at $s$ relative to $\mathbb{R}$. The closed interval $[\alpha,\beta]$ is an interval of which every point of $(\alpha,\beta)$ is an interior point (Basic Facts about Intervals of the Real Line and Their Interior Points §closed-interval); by claim 1 of Restriction Stability of Continuity and of the Derivative the restriction $g|_{[\alpha,\beta]}$ is continuous on $[\alpha,\beta]$, and by claim 2 there it is differentiable at every point of $(\alpha,\beta)$, with the derivative of $g$. By Mean Value Theorem on a Closed Real Interval there is $c\in(\alpha,\beta)$ with $g(\beta)-g(\alpha)=g'(c)(\beta-\alpha)$. Hence $|\eta(w^{j})-\eta(w^{j-1})|=|g(\beta)-g(\alpha)|=|g'(c)|(\beta-\alpha)\le D\,|z'_{j}-z_{j}|$, proving $(\ast)$.

Claim $(\ast\ast)$: for every $j\in[d]$, $|\eta(w^{j})-\eta(z)|\le j\,D\,\lVert z'-z\rVert$. For every $j\in[d]$, $|z'_{j}-z_{j}|\le\lVert z'-z\rVert$ by the coordinate bound, so $(\ast)$ gives $|\eta(w^{j})-\eta(w^{j-1})|\le D\lVert z'-z\rVert$; for $j=1$ this is $(\ast\ast)$, as $w^{0}=z$. If $(\ast\ast)$ holds for some $j$ with $j+1\in[d]$, then

the image of $j+1$ in $\mathbb{R}$ being that of $j$ plus $1$ by claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. By induction on $j$ (Natural Numbers), $(\ast\ast)$ holds for every $j\in[d]$; with $j=d$ and $w^{d}=z'$ it gives $|\eta(z)-\eta(z')|\le d\,D\,\lVert z-z'\rVert$.

For the rescaled kernel, $\eta_{\varepsilon}(z)-\eta_{\varepsilon}(z')=c_{\varepsilon}\bigl(\eta(\varepsilon^{-1}z)-\eta(\varepsilon^{-1}z')\bigr)$ and $\lVert\varepsilon^{-1}z-\varepsilon^{-1}z'\rVert=\lVert\varepsilon^{-1}(z-z')\rVert=\varepsilon^{-1}\lVert z-z'\rVert$ by homogeneity, so the bound just proved gives $|\eta_{\varepsilon}(z)-\eta_{\varepsilon}(z')|\le c_{\varepsilon}\,d\,D\,\varepsilon^{-1}\lVert z-z'\rVert=(\varepsilon^{-1})^{d+1}d\,D\,\lVert z-z'\rVert$.

(1g) Borel compositions. Let $m$ be a natural number and $u,v:\mathbb{R}^{m}\to\mathbb{R}^{d}$ Borel. By the componentwise criterion recorded in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps (claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets), the components $u_{i},v_{i}$ are Borel; the $i$th component of $w\mapsto v(w)-u(w)$ is $v_{i}-u_{i}$, which is Borel by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; so $w\mapsto v(w)-u(w)$ is Borel by the same criterion, and its composition with the Borel map $\eta_{\varepsilon}$ of (1b) is Borel, a composition of Borel maps being Borel (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps). We shall use this with maps built from the identity of $\mathbb{R}^{d}$ and constant maps, which are Borel by the same criterion, their components being coordinate projections (claims 1 and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets), respectively constants (claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), and with the coordinate projections of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections.

(1h) Unit mass of translates. Let $x\in\mathbb{R}^{d}$. By (1g) with $m=d$, $v$ the identity and $u$ constant equal to $x$, the function $y\mapsto\eta_{\varepsilon}(y-x)$ is Borel; it is nonnegative by (1a). Since $y+(-x)=y-x$, claim 2 of Translation and Reflection Invariance of Lebesgue Measure on $\mathbb{R}^n$, applied with $n=d$, $a=-x$ and the nonnegative measurable function $\eta_{\varepsilon}$, gives $\int_{\mathbb{R}^{d}}\eta_{\varepsilon}(y-x)\,\lambda_{d}(dy)=\int_{\mathbb{R}^{d}}\eta_{\varepsilon}\,d\lambda_{d}$ in $[0,\infty]$. For a nonnegative integrable function the integral in $[0,\infty]$ equals the real integral, its positive part being the function and its negative part $0$ (Integrable Function and the Lebesgue Integral); so by (1a) both sides equal $1$, and $y\mapsto\eta_{\varepsilon}(y-x)$ is integrable by the criterion of Measure Spaces and the Lebesgue Integral: Standing Notation §integral. This completes the proof of claim 1.

Step 2 (Claim 2). Let $D$ be as in claim 1, let $x,x'\in\mathbb{R}^{d}$, put $\delta=\lVert x-x'\rVert$, and let $f(y)=|\eta_{\varepsilon}(y-x)-\eta_{\varepsilon}(y-x')|$. By (1h) and claims 2 and 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, $f$ is Borel, and it is nonnegative. Put $K_{0}=\varepsilon^{-1}(2+2^{d}d\,\kappa_{d}D)$; here $0<\kappa_{d}$ by The Lebesgue Measure of a Closed Ball in $\mathbb{R}^n$ §constant, so $2^{d}d\,\kappa_{d}D\ge0$.

Since both kernels are nonnegative, $f(y)\le\eta_{\varepsilon}(y-x)+\eta_{\varepsilon}(y-x')$, so claim 1 of Linearity and Monotonicity of the Lebesgue Integral and (1h) give $\int_{\mathbb{R}^{d}}f\,d\lambda_{d}\le1+1=2$.

Case $\varepsilon\le\delta$. Then $2=2\varepsilon^{-1}\varepsilon\le2\varepsilon^{-1}\delta\le K_{0}\,\delta$.

Case $\delta<\varepsilon$. Let $B=\bar{B}(x,2\varepsilon)$; by The Lebesgue Measure of a Closed Ball in $\mathbb{R}^n$ §borel and The Lebesgue Measure of a Closed Ball in $\mathbb{R}^n$ §value it is Borel with $\lambda_{d}(B)=\kappa_{d}(2\varepsilon)^{d}$. If $y\notin B$, then $2\varepsilon<d_{E}(x,y)=\lVert y-x\rVert$, and the triangle inequality applied to $y-x=(y-x')+(x'-x)$, with $\lVert x'-x\rVert=\delta$, gives $\lVert y-x'\rVert\ge\lVert y-x\rVert-\delta>2\varepsilon-\varepsilon=\varepsilon$; since also $\varepsilon<2\varepsilon<\lVert y-x\rVert$, (1a) gives $\eta_{\varepsilon}(y-x)=\eta_{\varepsilon}(y-x')=0$, so $f(y)=0$. If $y\in B$, then, as $(y-x)-(y-x')=x'-x$ has norm $\delta$, (1f) gives $f(y)\le(\varepsilon^{-1})^{d+1}d\,D\,\delta$. Hence $f\le(\varepsilon^{-1})^{d+1}d\,D\,\delta\,\mathbf{1}_{B}$ pointwise, and by claim 1 of Linearity and Monotonicity of the Lebesgue Integral and The Integral of an Indicator Function is the Measure of the Set,

using $(\varepsilon^{-1})^{d+1}(2\varepsilon)^{d}=\varepsilon^{-1}\,2^{d}\,(\varepsilon^{-1}\varepsilon)^{d}=2^{d}\varepsilon^{-1}$. In both cases $\int f\,d\lambda_{d}\le K_{0}\,\delta$, which is claim 2.

Step 3 (Claim 3). Let $\mu\in\mathcal{P}(\mathbb{R}^{d})$, and fix a positive $S_{0}$ with $\eta\le S_{0}$ as in (1d). The function $\eta_{\varepsilon}$ is continuous by (1b) and satisfies $|\eta_{\varepsilon}(z)|=\eta_{\varepsilon}(z)\le c_{\varepsilon}S_{0}$ for all $z$ by (1d). Hence The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral §continuous, applied with $q=d$, $H=\eta_{\varepsilon}$ and $C=c_{\varepsilon}S_{0}$, shows: for every $y$ the function $x\mapsto\eta_{\varepsilon}(y-x)$ is Borel and integrable with respect to $\mu$, and it is bounded by $c_{\varepsilon}S_{0}$; the function $\eta_{\varepsilon}*\mu$, which is the function $H*\mu$ of that lemma, is continuous; and so it is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps.

Let $S$ be any real with $\eta\le S$. For $y\in\mathbb{R}^{d}$ we have $0\le\eta_{\varepsilon}(y-x)\le c_{\varepsilon}S$ for every $x$ by (1d). As $\mu$ is a Borel measure with $\mu(\mathbb{R}^{d})=1$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures), the constant functions $0$ and $c_{\varepsilon}S$ are integrable with integrals $0$ and $c_{\varepsilon}S$ by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space; the monotonicity statement in claim 2 of Linearity and Monotonicity of the Lebesgue Integral gives $0\le(\eta_{\varepsilon}*\mu)(y)\le c_{\varepsilon}S$.

Unit mass. Let $k:\mathbb{R}^{d+d}\to\mathbb{R}$, $k(w)=\eta_{\varepsilon}(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w))$, with the coordinate projections of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs; it is Borel by (1g) (with $m=d+d$, $u=\mathrm{pr}_{1}$, $v=\mathrm{pr}_{2}$) and nonnegative. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product the concatenation $\iota:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}^{d+d}$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{d})\otimes\mathcal{B}(\mathbb{R}^{d})$ and $\mathcal{B}(\mathbb{R}^{d+d})$, the latter being the Borel $\sigma$-algebra of $(\mathbb{R}^{d+d},d_{E})$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces); so $\hat{k}=k\circ\iota$ is measurable with respect to the product $\sigma$-algebra by claim 4 of Borel Measurability and Bounded Integration on a Metric Space, and $\hat{k}(x,y)=\eta_{\varepsilon}(y-x)$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections. The measure $\mu$ is finite and $\lambda_{d}$ is $\sigma$-finite by Lebesgue Measure on Euclidean Space is Sigma-Finite §sigma-finite, so the Tonelli statement of Tonelli and Fubini Theorems, applied to the measure spaces $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$ and $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\lambda_{d})$ and the function $\hat{k}$, gives

By (1h) the inner integral on the right is $1$ for every $x$, so the right side is $\int_{\mathbb{R}^{d}}\mathbf{1}_{\mathbb{R}^{d}}\,d\mu=\mu(\mathbb{R}^{d})=1$ by The Integral of an Indicator Function is the Measure of the Set. On the left, the inner integral of the nonnegative integrable function $x\mapsto\eta_{\varepsilon}(y-x)$ equals its real integral $(\eta_{\varepsilon}*\mu)(y)$, as in (1h). Hence the nonnegative Borel function $\eta_{\varepsilon}*\mu$ has integral $1$ in $[0,\infty]$; by the criterion of Measure Spaces and the Lebesgue Integral: Standing Notation §integral it is integrable with respect to $\lambda_{d}$, and its real integral is $1$, as in (1h). This proves claim 3.

Step 4 (Claim 4). Let $\mu$, $L$ and $\Phi$ be as in claim 4. By claim 3, $h=\eta_{\varepsilon}*\mu$ is a Borel, nonnegative function, integrable with respect to $\lambda_{d}$ with integral $1$. The composition clause Supporting Lines, Composition and Jensen's Inequality for a Convex Lipschitz Integrand §composition, applied to the measure space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\lambda_{d})$, whose measurable real functions are the Borel ones (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), and to $h$, shows that $\Phi\circ h$ is Borel, that $0\le\Phi(h(y))\le L\,h(y)$ for every $y$, and that $\Phi\circ h$ is integrable with $0\le\int\Phi\circ h\,d\lambda_{d}\le L\int h\,d\lambda_{d}=L$. This proves claim 4.