Proof of The Lifted Free Score: Norm, Weak Identity and the Second-Moment Identity

Each result cited is universally quantified over the data in its own statement. The identifications of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative are in force, in particular $\lVert x\rVert=|x|$ for $x\in\mathbb{R}$ (One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative (i)), and $\iota:\mathbb{N}\to\mathbb{R}$ is the canonical map of The Canonical Map from the Natural Numbers to a Field. Convergence of a sequence of real numbers is that of Limit of a Sequence of Real Numbers, read as stipulated in the preamble of Convergence, Limits and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space, which is Complete: for a sequence $(a_{m})_{m\in\mathbb{N}}$ the indices $n$ for which $a_{n}$ is defined are exactly those of the form $n=\iota(m)$ with $m\in\mathbb{N}$, and for these the condition "$n\ge N$" reads $\iota(N)\le\iota(m)$. For every $\nu\in\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$ we have $\nu\in\mathcal{P}_{2}(\mathbb{R})$ by Finite Free Fisher Information, the Free Score and the Free Fisher Information of a Probability Measure on the Real Line (i), and $\Xi_{\nu}\in T_{\nu}\subseteq L^{2}(\nu;\mathbb{R})$ by Finite Free Fisher Information, the Free Score and the Free Fisher Information of a Probability Measure on the Real Line (ii) and Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients (i); $L^{2}(\nu;\mathbb{R})$ is a real Hilbert space with distance $d_{\nu}$ by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (ii).

Step 1: claim 1. Suppose that the law $\mu=\mathcal{L}(X)$ lies in $\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$. By Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity (i), applied with $d=1$, $\lVert\Xi_{\mu}\circ X\rVert_{L^{2}}=\lVert\Xi_{\mu}\rVert_{\mu}$, so $\lVert\Xi_{\mu}\circ X\rVert_{L^{2}}^{2}=\lVert\Xi_{\mu}\rVert_{\mu}^{2}=\Phi^{*}(\mu)$ by Finite Free Fisher Information, the Free Score and the Free Fisher Information of a Probability Measure on the Real Line (iii). Let $\psi\in C_{c}^{\infty}(\mathbb{R})$; then $\psi'=\nabla\psi$ by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative (ii), and its class lies in $L^{2}(\mu;\mathbb{R})$ by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation (iii), so Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity (i) gives $\langle\Xi_{\mu}\circ X,\psi'\circ X\rangle_{L^{2}}=\langle\Xi_{\mu},\nabla\psi\rangle_{\mu}$, which equals $\int_{\mathbb{R}^{2}}F_{\psi}\,d(\mu\boxtimes\mu)$ by Finite Free Fisher Information, the Free Score and the Free Fisher Information of a Probability Measure on the Real Line (ii).

Step 2: the limit of the difference-quotient integrals. Let $\nu\in\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$, and for $n\in\mathbb{N}$ let $\psi_{n}$ be the second-moment test function of the preamble of Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients, applied with $\nu$ in place of $\mu$ and with $d=1$: $\psi_{n}$ is the function $\psi_{R}$ of Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball for the dimension $q=1$ and $R=\iota(n)$, positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and it is a test function, as recorded there. Fix a nonnegative real number $M$ as in Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball (ii) for $q=1$, not depending on $R$; since $qM=\iota(1)M=M$ by claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, that clause gives $|\Delta\psi_{n}(t)|\le M$ for all $n\in\mathbb{N}$ and $t\in\mathbb{R}$. Hence, by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative (iii), each $F_{\psi_{n}}$ is Borel and $|F_{\psi_{n}}(z)|\le M$ for all $z\in\mathbb{R}^{2}$ and $n\in\mathbb{N}$.

Pointwise limit. Fix $(x,y)\in\mathbb{R}^{2}$. By claim 1 of The Archimedean Property of the Real Numbers there are $n_{1},n_{2}\in\mathbb{N}$ with $|x|<\iota(n_{1})$ and $|y|<\iota(n_{2})$; put $n_{0}=n_{1}+n_{2}$. By claim 6 of Properties of the Order on the Natural Numbers, $n_{1}<n_{1}+n_{2}=n_{0}$, and $n_{2}<n_{2}+n_{1}=n_{0}$ by the same claim and claim 4 of Arithmetic of Addition on the Natural Numbers; so $\iota(n_{1})<\iota(n_{0})$ and $\iota(n_{2})<\iota(n_{0})$ by claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and $|x|<\iota(n_{0})$, $|y|<\iota(n_{0})$ by claim 2 of Elementary Order Arithmetic in an Ordered Field (a strict inequality implying the weak one). Now let $m\in\mathbb{N}$ satisfy $\iota(n_{0})\le\iota(m)$. Then $\lVert x\rVert=|x|<\iota(m)$ and $\lVert y\rVert<\iota(m)$ by claim 2 of Elementary Order Arithmetic in an Ordered Field, so Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball (iii), applied with the positive number $R=\iota(m)$, gives $\partial_{1}\psi_{m}(x)=x_{1}=x$, $\partial_{1}\psi_{m}(y)=y$ and $\Delta\psi_{m}(x)=q$, read as $\iota(1)$ by the preamble of that lemma, which is $1$ by claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field; and $\psi_{m}'=\partial_{1}\psi_{m}$ by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative (ii). If $x\ne y$, then $F_{\psi_{m}}(x,y)=(x-y)(x-y)^{-1}=1$; if $x=y$, then $F_{\psi_{m}}(x,x)=\Delta\psi_{m}(x)=1$. Thus, for every real $\varepsilon>0$, $|F_{\psi_{m}}(x,y)-1|=0<\varepsilon$ for every $m\in\mathbb{N}$ with $\iota(n_{0})\le\iota(m)$, that is, the real sequence $(F_{\psi_{n}}(x,y))_{n\in\mathbb{N}}$ converges to $1$.

Dominated convergence. By Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v) and Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii), $\nu\boxtimes\nu\in\mathcal{P}(\mathbb{R}^{2})$ is a Borel measure on the metric space $(\mathbb{R}^{2},d_{E})$ with $(\nu\boxtimes\nu)(\mathbb{R}^{2})=1$, and $\mathcal{B}(\mathbb{R}^{2})$ is its Borel $\sigma$-algebra (Probability Measures on Euclidean Space and Random Vectors: Standing Notation (i)); so by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space the constant functions with values $M$ and $1$ on $\mathbb{R}^{2}$ are integrable with respect to $\nu\boxtimes\nu$, with $\int_{\mathbb{R}^{2}}1\,d(\nu\boxtimes\nu)=1\cdot(\nu\boxtimes\nu)(\mathbb{R}^{2})=1$. Dominated Convergence Theorem, applied on the measure space $(\mathbb{R}^{2},\mathcal{B}(\mathbb{R}^{2}),\nu\boxtimes\nu)$ to the sequence $f_{n}=F_{\psi_{n}}$ of Borel functions, the limit function $f=1$ and the dominating function $g=M$, yields

$$\int_{\mathbb{R}^{2}}F_{\psi_{n}}\,d(\nu\boxtimes\nu)\to\int_{\mathbb{R}^{2}}1\,d(\nu\boxtimes\nu)=1\qquad(n\to\infty).$$

Step 3: claim 2. Let $\nu$ and $\psi_{n}$ be as in Step 2. By Finite Free Fisher Information, the Free Score and the Free Fisher Information of a Probability Measure on the Real Line (ii) applied to $\psi_{n}$, the real sequence $a_{n}=\langle\Xi_{\nu},\nabla\psi_{n}\rangle_{\nu}=\int_{\mathbb{R}^{2}}F_{\psi_{n}}\,d(\nu\boxtimes\nu)$ converges to $1$ by Step 2. By the symmetry of the inner product (Real Inner Product Spaces (i)), $a_{n}=\langle\nabla\psi_{n},\Xi_{\nu}\rangle_{\nu}$. Since $\nabla\psi_{n}\to\mathrm{id}$ in the metric space $(L^{2}(\nu;\mathbb{R}),d_{\nu})$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients (ii), the last assertion of The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (v), applied in the real inner product space $L^{2}(\nu;\mathbb{R})$ with $z=\Xi_{\nu}$, shows that $(a_{n})$ converges to $\langle\mathrm{id},\Xi_{\nu}\rangle_{\nu}=\langle\Xi_{\nu},\mathrm{id}\rangle_{\nu}$. By claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences, $\langle\Xi_{\nu},\mathrm{id}\rangle_{\nu}=1$. If $\mu\in\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$, this applies to $\nu=\mu$, and by Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity (i), $\langle\Xi_{\mu}\circ X,X\rangle_{L^{2}}=\langle\Xi_{\mu}\circ X,\mathrm{id}\circ X\rangle_{L^{2}}=\langle\Xi_{\mu},\mathrm{id}\rangle_{\mu}=1$.