Proof of The Lift of a Linear Functional of the Measure: Integrability, L-Gradient, Lipschitz Gradient Map and Translation Laplacian

Each result cited is universally quantified over the data in its own statement. Write $L^{2}$ for $L^{2}(\Omega;\mathbb{R}^{d})$; the natural number $d$ is read in $\mathbb{R}$ through the canonical map $\iota$ of The Canonical Map from the Natural Numbers to a Field, and we write $d$ for $\iota(d)$ inside real expressions, as in the statement. Elements of $L^{2}$ are handled through representatives, again written with the same letter, as the convention of The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §space allows; the operations on classes are those of The Space of Square-Integrable Random Vectors §classes. For $x,h\in\mathbb{R}^{d}$ the segment $\{x+\tau h:0\le\tau\le1\}$ lies in $\mathbb{R}^{d}$, which is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and $|h|=d_{E}(x,x+h)=\lVert h\rVert$ in the notation of Multivariate Taylor Expansion with Uniform Second-Order Remainder by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$. Since $f$ is of class $C^{2}$, each $\partial_{i}f$ is of class $C^{1}$ with partial derivatives $\partial_{j}\partial_{i}f$, by clause 2 of C^k Maps on a Euclidean Open Set. We record four pointwise bounds, valid for all $x,h\in\mathbb{R}^{d}$.

(T1) $|f(x+h)-f(x)|\le\sqrt{d}\,M_{1}\lVert h\rVert$, by claim (i) of Multivariate Taylor Expansion with Uniform Second-Order Remainder applied to $f$ with the bound $M_{1}$.

(T2) $\bigl|f(x+h)-f(x)-Df(x)\cdot h\bigr|\le\tfrac12\,d\,M_{2}\lVert h\rVert^{2}$, by claim (ii) there applied to $f$ with the bound $M_{2}$, since $\sum_{i=1}^{d}\partial_{i}f(x)h_{i}=Df(x)\cdot h$ by Gradient of a Real-Valued Function on a Euclidean Open Set and Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$.

(T3) $|\partial_{i}f(x+h)-\partial_{i}f(x)|\le\sqrt{d}\,M_{2}\lVert h\rVert$ for every $i\in[d]$, by claim (i) there applied to $\partial_{i}f$ with the bound $M_{2}$.

(T4) $\lVert Df(x+h)-Df(x)\rVert\le d\,M_{2}\lVert h\rVert$. Indeed, by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, clause 1 of Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$ and Gradient of a Real-Valued Function on a Euclidean Open Set, $\lVert Df(x+h)-Df(x)\rVert^{2}=\sum_{i=1}^{d}(\partial_{i}f(x+h)-\partial_{i}f(x))^{2}$; each summand is at most $(\sqrt{d}M_{2}\lVert h\rVert)^{2}=d\,M_{2}^{2}\lVert h\rVert^{2}$ by (T3), claim 1 of Nonnegativity of Squares in an Ordered Field and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field (the square root being that of Existence and Uniqueness of the Nonnegative Square Root); summing, by claims 2 and 5 of Properties of Finite Sums applied to the nonnegative differences, then claim 3 there and The Canonical Map from the Natural Numbers to a Field for the constant sum, $\lVert Df(x+h)-Df(x)\rVert^{2}\le d\cdot d\,M_{2}^{2}\lVert h\rVert^{2}=(d\,M_{2}\lVert h\rVert)^{2}$, and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives (T4), both sides being nonnegative.

(G) Also, by (T1) with $x=0_{\mathbb{R}^{d}}$ and claim 5 of Properties of the Absolute Value in an Ordered Field, for every $x\in\mathbb{R}^{d}$

the second step because $\lVert x\rVert\le1+\lVert x\rVert^{2}$: if $\lVert x\rVert\le1$ this holds as $0\le\lVert x\rVert^{2}$ (claim 2 of Nonnegativity of Squares in an Ordered Field), and if $1<\lVert x\rVert$ then $\lVert x\rVert=\lVert x\rVert\cdot1<\lVert x\rVert\cdot\lVert x\rVert$ by claim 10 of Elementary Order Arithmetic in an Ordered Field, so $\lVert x\rVert\le\lVert x\rVert^{2}\le1+\lVert x\rVert^{2}$; then multiply by the nonnegative $\sqrt{d}M_{1}$ (claim 5 of Elementary Arithmetic in an Ordered Field).

Claim 1. Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, a probability measure on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. The function $g(x)=|f(0_{\mathbb{R}^{d}})|+\sqrt{d}M_{1}+\sqrt{d}M_{1}\lVert x\rVert^{2}$ is integrable with respect to $\mu$: the constant is integrable by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space ($\mu$ being a Borel measure with $\mu(\mathbb{R}^{d})=1$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures), the nonnegative Borel function $x\mapsto\lVert x\rVert^{2}$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs) has integral $M_{2}(\mu)<\infty$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment and The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space, hence is integrable by Integrable Function and the Lebesgue Integral (the two readings of its integral agreeing by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures), and linear combinations of integrable functions are integrable by claim 2 of Linearity and Monotonicity of the Lebesgue Integral. Since $f$ is Borel, $|f|$ is Borel by claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and $|f|\le g$ pointwise by (G); so $\int|f|\,d\mu\le\int g\,d\mu<\infty$ by the monotonicity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and $f$ is integrable with respect to $\mu$ by Integrable Function and the Lebesgue Integral. Thus $u$ is defined. For $X\in L^{2}$ and a representative of it, $\mathcal{L}(X)\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §law-map, so $f$ is integrable with respect to $\mathcal{L}(X)$, whence $f\circ X$ is integrable and $\mathbb{E}[f\circ X]=\int f\,d\mathcal{L}(X)=u(\mathcal{L}(X))=U(X)$ by Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §expectation and The Lift of a Function on the Wasserstein Space to the Space of Square-Integrable Random Vectors §lift.

Claim 2. Square-integrability and the class. For every $x\in\mathbb{R}^{d}$, $\lVert Df(x)\rVert^{2}=\sum_{i=1}^{d}\partial_{i}f(x)^{2}\le\sum_{i=1}^{d}M_{1}^{2}=d\,M_{1}^{2}$, by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, the hypothesis $|\partial_{i}f|\le M_{1}$ with claim 1 of Nonnegativity of Squares in an Ordered Field and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, claims 2 and 5 of Properties of Finite Sums, and the constant sum as in (T4). Hence, for a random vector $X$, $\mathbb{E}[\lVert Df\circ X\rVert^{2}]\le\mathbb{E}[d\,M_{1}^{2}]=d\,M_{1}^{2}P(\Omega)=d\,M_{1}^{2}<\infty$ by the monotonicity and homogeneity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral and The Integral of an Indicator Function is the Measure of the Set (the constant being $d\,M_{1}^{2}\mathbf{1}_{\Omega}$), so $Df\circ X$ is square-integrable by The Space of Square-Integrable Random Vectors §space. If $X'$ is another representative, $P(X=X')=1$ by The Space of Square-Integrable Random Vectors §classes; the event $\{Df\circ X=Df\circ X'\}$ (an event by Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §almost-sure) contains $\{X=X'\}$, so it has probability $1$ by the monotonicity of $P$ (claim 2 of Basic Properties of a Measure) and $P(\Omega)=1$, whence $Df\circ X$ and $Df\circ X'$ have the same class by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §almost-sure.

Differentiability. Fix $X\in L^{2}$ and let $Z\in L^{2}$, with representatives $X,Z$; then $X+Z$ is a representative of the class $X+Z$. Applying (T2) at each $\omega$ with $x=X(\omega)$ and $h=Z(\omega)$,

where $(Df\circ X)\cdot Z$ and $\lVert Z\rVert^{2}$ are the random variables of Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §composition. The three random variables on the left are integrable: $f\circ(X+Z)$ and $f\circ X$ by claim 1, and $(Df\circ X)\cdot Z$ by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §operations, with $\mathbb{E}[(Df\circ X)\cdot Z]=\langle Df\circ X,Z\rangle_{L^{2}}$ by The Space of Square-Integrable Random Vectors §inner-product; and $\mathbb{E}[\lVert Z\rVert^{2}]=\lVert Z\rVert_{L^{2}}^{2}$ by The Space of Square-Integrable Random Vectors §inner-product and Existence and Uniqueness of the Nonnegative Square Root. By claim 2 of Linearity and Monotonicity of the Lebesgue Integral (linearity, $|\int\cdot|\le\int|\cdot|$ and monotonicity, the absolute value of an integrable function being measurable by claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and integrable by Integrable Function and the Lebesgue Integral), together with claim 1,

Let $\varepsilon>0$. If $d\,M_{2}=0$, the right-hand side is $0\le\varepsilon\lVert Z\rVert_{L^{2}}$ for every $Z$ (claim 1 of Zero Products and Elementary Identities in a Field, $\lVert Z\rVert_{L^{2}}\ge0$ by Real Inner Product Space §norm). Otherwise $M_{2}\ne0$, so $0<M_{2}$ as $M_{2}$ is nonnegative, and $d\,M_{2}>0$ by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field and claim 5 of Elementary Order Arithmetic in an Ordered Field; we put $\delta=2\varepsilon\,(d\,M_{2})^{-1}>0$ (claims 7 and 5 of Elementary Order Arithmetic in an Ordered Field); for $\lVert Z\rVert_{L^{2}}<\delta$ we get $\tfrac12\,d\,M_{2}\lVert Z\rVert_{L^{2}}^{2}=\tfrac12\,d\,M_{2}\lVert Z\rVert_{L^{2}}\cdot\lVert Z\rVert_{L^{2}}\le\tfrac12\,d\,M_{2}\,\delta\,\lVert Z\rVert_{L^{2}}=\varepsilon\lVert Z\rVert_{L^{2}}$ by claim 5 of Elementary Arithmetic in an Ordered Field. In either case every $Z$ with $\lVert Z\rVert_{L^{2}}<\delta$ (any $\delta>0$ in the first case) satisfies $X+Z\in L^{2}$ and the inequality of Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §differentiable with $p=Df\circ X$; so $U$ is differentiable at $X$ with gradient $DU(X)=Df\circ X$ (Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §gradient). As $X$ was arbitrary, $U$ is differentiable on $L^{2}$ (Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §on-set), that is, $u$ is $L$-differentiable with $L$-gradient $Df\circ X$ at $X$ by L-Differentiability of a Function on the Wasserstein Space via the Fréchet Derivative of Its Lift §differentiable.

Claim 3. Let $X,Y\in L^{2}$ with representatives $X,Y$. By claim 2, $DU(X)-DU(Y)$ is the class of $Df\circ X-Df\circ Y$ (The Space of Square-Integrable Random Vectors §classes and Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §vector-space), and (T4) at each $\omega$ with $x=Y(\omega)$, $h=X(\omega)-Y(\omega)$ gives $\lVert Df\circ X-Df\circ Y\rVert\le d\,M_{2}\lVert X-Y\rVert$ pointwise, hence $\lVert Df\circ X-Df\circ Y\rVert^{2}\le d^{2}M_{2}^{2}\lVert X-Y\rVert^{2}$ pointwise by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Taking expectations (monotonicity and homogeneity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral) and using The Space of Square-Integrable Random Vectors §inner-product with Existence and Uniqueness of the Nonnegative Square Root on both sides,

and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives the asserted inequality, both sides being nonnegative. The gradient map is therefore continuous on $L^{2}$: given $Y$ and $\varepsilon>0$, $\delta=\varepsilon(d\,M_{2}+1)^{-1}$ gives $\lVert DU(X)-DU(Y)\rVert_{L^{2}}\le d\,M_{2}\lVert X-Y\rVert_{L^{2}}<\varepsilon$ whenever $\lVert X-Y\rVert_{L^{2}}<\delta$ (claim 5 of Elementary Arithmetic in an Ordered Field, claim 10 of Elementary Order Arithmetic in an Ordered Field and $d\,M_{2}<d\,M_{2}+1$), which is Continuous Map Between Metric Spaces for the metric $d_{L^{2}}$. Hence $U\in C^{1}(L^{2})$ by The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c1, and $u$ is continuously $L$-differentiable by L-Differentiability of a Function on the Wasserstein Space via the Fréchet Derivative of Its Lift §c1.

Claim 4. Fix $X\in L^{2}$ with a representative $X$, and let $\phi_{X}(a)=U(X+c_{a})$ be the function of The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors. For $a\in\mathbb{R}^{d}$ the map $X+a:\omega\mapsto X(\omega)+a$ is a representative of $X+c_{a}$ (The Space of Square-Integrable Random Vectors §classes, The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §constants), so by claim 1

For $i\in[d]$ let $e_{i}\in\mathbb{R}^{d}$ have $i$th component $1$ and every other component $0$; the point with $i$th component $a_{i}+h$ and the other components those of $a$ is $a+he_{i}$, since both points have the same components (claims 1 and 2 of Euclidean Points as Tuples of Real Numbers, the components of $a+he_{i}$ being $a_{k}+h(e_{i})_{k}$ by Sum of Points of $\mathbb{R}^n$ and Scalar Multiple of a Point of $\mathbb{R}^n$).

Step A: first partial derivatives. Let $g:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{1}$ with $|\partial_{i}g|\le K$ on $\mathbb{R}^{d}$ for every $i\in[d]$ and some nonnegative $K$, and suppose that $g\circ(X+b)$ is integrable for every $b\in\mathbb{R}^{d}$. Put $\psi(a)=\mathbb{E}[g\circ(X+a)]$. We show that for every $a\in\mathbb{R}^{d}$ and $i\in[d]$ the partial derivative $\partial_{i}\psi(a)$ exists and equals $\mathbb{E}[\partial_{i}g\circ(X+a)]$. Let $J=(-1,1)$, an interval all of whose points are interior by An Open Interval is an Interval All of Whose Points Are Interior, and let $F:J\times\Omega\to\mathbb{R}$, $F(t,\omega)=g(X(\omega)+a+te_{i})$. Condition (i) of Differentiation under the Integral Sign holds by hypothesis with $b=a+te_{i}$. For (ii), fix $\omega$ and put $x=X(\omega)+a$: by claim 2 of Derivatives Along a Segment for C^1 Functions on a Euclidean Open Set with $h=e_{i}$, the function $t\mapsto g(x+te_{i})$ is differentiable at every point $t$ of $J$ with derivative $\sum_{k=1}^{d}\partial_{k}g(x+te_{i})(e_{i})_{k}=\partial_{i}g(x+te_{i})$, the sum having a single possibly nonzero summand (claim 7 of Properties of Finite Sums), so $D_{1}F(t,\omega)=\partial_{i}g(X(\omega)+a+te_{i})$. For (iii), $|D_{1}F(t,\omega)|\le K$, and the constant $K\mathbf{1}_{\Omega}$ is integrable on $(\Omega,\mathcal{F},P)$ with integral $K$, by The Integral of an Indicator Function is the Measure of the Set, the homogeneity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral and Integrable Function and the Lebesgue Integral. The theorem gives that $t\mapsto\mathbb{E}[g\circ(X+a+te_{i})]=\psi(a+te_{i})$ is differentiable on $J$ with derivative $\mathbb{E}[\partial_{i}g\circ(X+a+te_{i})]$ at $t$; at $t=0$ this reads: for every $\varepsilon>0$ there is $\delta>0$ such that every $h$ with $0<|h|<\delta$ and $h\in J$ satisfies $\bigl|\frac{\psi(a+he_{i})-\psi(a)}{h}-\mathbb{E}[\partial_{i}g\circ(X+a)]\bigr|<\varepsilon$ (Derivative at an Interior Point). Taking $\delta\le1$, the restriction $h\in J$ is automatic, and since $a+he_{i}$ is the point with $i$th component $a_{i}+h$, this is exactly the statement of Partial Derivative on a Euclidean Open Set that $\partial_{i}\psi(a)$ exists with value $\mathbb{E}[\partial_{i}g\circ(X+a)]$.

Step B: continuity of the averaged functions. Let $g:\mathbb{R}^{d}\to\mathbb{R}$ be continuous at every point in the Euclidean sense and bounded by a nonnegative constant $K$, and put $\psi(a)=\mathbb{E}[g\circ(X+a)]$, defined since $g$ is Borel (claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), so that $g\circ(X+a)$ is a random variable by Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §composition, bounded by $K$, hence integrable by the monotonicity and homogeneity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral against the constant $K\mathbf{1}_{\Omega}$, whose integral is $K$ by The Integral of an Indicator Function is the Measure of the Set. We show $\psi$ is continuous at every point in the Euclidean sense. By claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions and Continuity Between Metric Spaces is Equivalent to Sequential Continuity §criterion it suffices to show: if $(a_{m})_{m\in\mathbb{N}}$ converges to $a$ in $(\mathbb{R}^{d},d_{E})$ then $\psi(a_{m})\to\psi(a)$. For each $\omega$, $d_{E}(X(\omega)+a_{m},X(\omega)+a)=\lVert a_{m}-a\rVert=d_{E}(a_{m},a)$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, so $X(\omega)+a_{m}\to X(\omega)+a$ in $(\mathbb{R}^{d},d_{E})$ (Convergent Sequence in a Metric Space), and $g(X(\omega)+a_{m})\to g(X(\omega)+a)$ by claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions and Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential. The functions $g\circ(X+a_{m})$ are measurable and bounded by the integrable constant $K$, so Dominated Convergence Theorem gives $\psi(a_{m})=\int g\circ(X+a_{m})\,dP\to\int g\circ(X+a)\,dP=\psi(a)$.

Step C: assembly. By Step A with $g=f$, $K=M_{1}$ (integrability by claim 1), $\partial_{i}\phi_{X}(a)=\mathbb{E}[\partial_{i}f\circ(X+a)]$ for all $a$ and $i$. By Step A with $g=\partial_{i}f$, $K=M_{2}$ (of class $C^{1}$ with partials $\partial_{j}\partial_{i}f$; $\partial_{i}f\circ(X+b)$ integrable as in Step B, being bounded by $M_{1}$), $\partial_{j}\partial_{i}\phi_{X}(a)=\mathbb{E}[\partial_{j}\partial_{i}f\circ(X+a)]$ for all $a$ and $i,j$. By Step B with $g=\partial_{i}f$, $K=M_{1}$ and with $g=\partial_{j}\partial_{i}f$, $K=M_{2}$ (continuous at every point by Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, as recorded in the statement), the functions $\partial_{i}\phi_{X}$ and $\partial_{j}\partial_{i}\phi_{X}$ are continuous at every point. Finally $\phi_{X}$ itself is continuous at every point: $U$ is continuous on $L^{2}$ by Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space §continuous (it is differentiable by Claim 2 above), and $\lVert(X+c_{a})-(X+c_{b})\rVert_{L^{2}}=\lVert c_{a-b}\rVert_{L^{2}}=\lVert a-b\rVert$ by The Space of Square-Integrable Random Vectors §classes and The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §constants, so for $\varepsilon>0$ the $\delta$ of the continuity of $U$ at $X+c_{b}$ gives $|\phi_{X}(a)-\phi_{X}(b)|<\varepsilon$ whenever $\lVert a-b\rVert<\delta$, which is continuity at $b$ in the sense of Continuous Map Between Metric Spaces for $d_{E}$ and $d_{\mathbb{R}}$, hence in the Euclidean sense by claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions. Therefore $\phi_{X}$ is of class $C^{1}$ by clause 1 of C^k Maps on a Euclidean Open Set, each $\partial_{i}\phi_{X}$ is of class $C^{1}$ by the same clause, and $\phi_{X}$ is of class $C^{2}$ by clause 2 there. So $U$ is twice continuously differentiable along translations at $X$ (The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors §translations), hence on the whole space as $X$ was arbitrary (The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors §on-space), and by The Translation Laplacian of a Function on the Space of Square-Integrable Random Vectors §laplacian, The Laplacian of a Twice Continuously Differentiable Function §laplacian and $X+0_{\mathbb{R}^{d}}=X$,

the third equality by Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §integrable and Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear, each summand being integrable as in Step B. Lastly $\Delta f$ is Borel and bounded by $d\,M_{2}$ (claims 2 and 5 of Properties of Finite Sums and the constant sum), hence integrable with respect to $\mathcal{L}(X)$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, and $\mathbb{E}[\Delta f\circ X]=\int\Delta f\,d\mathcal{L}(X)$ by Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §expectation. The class of $X$ being arbitrary and $\mathcal{L}(X)$ being the same for all representatives by The Space of Square-Integrable Random Vectors §law, the identity holds for every representative.