Proof of Law of Large Numbers for Empirical Measures in the Wasserstein Distance

Each result cited is universally quantified over the data in its own statement.

Conventions. Fix $q\in\mathbb{N}$ and $\nu\in\mathcal{P}_{2}(\mathbb{R}^{q})$ as in the statement. By Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation we have $1\le q$, so for every $N\in\mathbb{N}$ also $1\le qN$, and Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound, in force by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §background, applies in dimension $q$. For $N\in\mathbb{N}$ we write $\iota(N)$ for the image of $N$ under the canonical map $\iota:\mathbb{N}\to\mathbb{R}$ of The Canonical Map from the Natural Numbers to a Field; this is the real number written $N$ in the displayed formulas of the results cited below. By claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, $0<\iota(N)$, the inverse $\iota(N)^{-1}$ exists and $0<\iota(N)^{-1}$. We write $P_{N}=\nu^{\otimes N}\in\mathcal{P}(\mathbb{R}^{qN})$ and $f_{N}(x)=W_{2}(\mu^{N}_{x},\nu)$ for $x\in\mathbb{R}^{qN}$. The triple $(\mathbb{R}^{qN},\mathcal{B}(\mathbb{R}^{qN}),P_{N})$ is a probability space in the sense of Probability Space, Event, and Random Variable, a Borel map $\mathbb{R}^{qN}\to\mathbb{R}$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps) is a random variable on it, and for a nonnegative random variable $V$ on it the expectation is $\mathbb{E}[V]=\int_{\mathbb{R}^{qN}}V\,dP_{N}$ by Expectation, Variance, and Moments; square-integrable random variables and the norm $\lVert V\rVert_{2}$, the nonnegative square root of $\mathbb{E}[V^{2}]$, are those of Square-Integrable Random Variables and the Mean-Square Inner Product. Two elementary facts are used repeatedly.

(C) Let $Q$ be a probability measure on a Euclidean space $E$ and let $c\in\mathbb{R}$ with $0\le c$. The constant function with value $c$ on $E$ is Borel by claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; it is a nonnegative simple function whose only value is $c$, attained on $E$, so its integral in the sense of Simple Function and Its Integral is $c\,Q(E)=c$, and by the last sentence of Lebesgue Integral of a Nonnegative Measurable Function this is also its integral as a nonnegative measurable function.

(S) Let $a,b\in\mathbb{R}$ with $0\le a$ and $0\le b$. Then $a\le b$ if and only if $a^{2}\le b^{2}$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field), and $a<b$ if and only if $a^{2}<b^{2}$ (claim 1 there). In particular, for $0\le s$ the nonnegative square root of $s$ given by Existence and Uniqueness of the Nonnegative Square Root is at most $b$ whenever $s\le b^{2}$.

Step 1 (Clause 1: integrability). Fix $N\in\mathbb{N}$. By Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §distance, applied with our $\nu\in\mathcal{P}_{2}(\mathbb{R}^{q})$, the function $f_{N}$ on $\mathbb{R}^{qN}$ is Borel with values in $[0,\infty)$, hence $f_{N}^{2}=f_{N}f_{N}$ is Borel by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, with nonnegative values. Let $x\in\mathbb{R}^{qN}$. By Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §moment, $\mu^{N}_{x}\in\mathcal{P}_{2}(\mathbb{R}^{q})$ and $\iota(N)M_{2}(\mu^{N}_{x})=\lVert x\rVert^{2}$; multiplying by $\iota(N)^{-1}$ gives $M_{2}(\mu^{N}_{x})=\iota(N)^{-1}\lVert x\rVert^{2}$. By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §product, $\mu^{N}_{x}\boxtimes\nu\in\Pi(\mu^{N}_{x},\nu)$; the inequality $W_{2}(\mu,\nu)^{2}\le I(\pi)$ for every coupling, recorded in The Quadratic Wasserstein Distance on Euclidean Space §distance, and the bound on the cost of a coupling of two members of $\mathcal{P}_{2}(\mathbb{R}^{q})$ in Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §cost-finite give

The function $x\mapsto\lVert x\rVert^{2}$ on $\mathbb{R}^{qN}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, and the constant $2M_{2}(\nu)$ is a nonnegative real number because $M_{2}(\nu)<\infty$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space. By the additivity, homogeneity and monotonicity of the integral of nonnegative measurable functions (claim 1 of Linearity and Monotonicity of the Lebesgue Integral),

The first integral is the second moment $M_{2}(P_{N})$ of The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment in dimension $qN$, which equals $\iota(N)M_{2}(\nu)$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments; the second equals $2M_{2}(\nu)$ by (C). Hence $\int_{\mathbb{R}^{qN}}f_{N}^{2}\,dP_{N}\le2M_{2}(\nu)+2M_{2}(\nu)<\infty$. As $f_{N}^{2}$ is nonnegative it equals $|f_{N}^{2}|$, so $f_{N}^{2}$ is integrable with respect to $P_{N}$ by the last sentence of Integrable Function and the Lebesgue Integral. This proves clause 1. Its integral is a nonnegative real number, and $\phi_{N}(\nu)$ is its nonnegative square root; thus $0\le\phi_{N}(\nu)$ and $\phi_{N}(\nu)^{2}=\int_{\mathbb{R}^{qN}}f_{N}^{2}\,dP_{N}$. In the language of the conventions, $f_{N}$ is a square-integrable random variable on $(\mathbb{R}^{qN},\mathcal{B}(\mathbb{R}^{qN}),P_{N})$ and $\phi_{N}(\nu)=\lVert f_{N}\rVert_{2}$.

Step 2 (Clause 2: the tolerance). To verify Limit of a Sequence of Real Numbers for the limit $0$, let $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ be given. The objects below are chosen in this order: first $\eta$ (Step 2), then $T$ (Step 3), then $M$, $z_{1},\dots,z_{M}$, $F$, $D$ and $A_{1},\dots,A_{M}$ (Step 4), then $m$, $K$, $\beta$ and $N_{0}$ (Step 5); each depends only on $\varepsilon$, $\nu$ and the objects chosen before it, and none depends on $N$. Put $\eta=(\varepsilon\cdot2^{-1})\cdot2^{-1}$. By claim 8 of Elementary Order Arithmetic in an Ordered Field, applied to $\varepsilon$ and then to $\varepsilon\cdot2^{-1}$, we have $0<\varepsilon\cdot2^{-1}$, $0<\eta$, $\eta+\eta=\varepsilon\cdot2^{-1}$ and $\varepsilon\cdot2^{-1}+\varepsilon\cdot2^{-1}=\varepsilon$, hence $\eta+\eta+\eta+\eta=\varepsilon$. Since $0<\eta$, claim 1 of Elementary Order Arithmetic in an Ordered Field (adding $\eta+\eta+\eta$ to both sides of $0<\eta$) gives $\eta+\eta+\eta<\varepsilon$.

Step 3 (Quantisation). By the second sentence of Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §quantisation, applied in dimension $q$ to $\nu\in\mathcal{P}_{2}(\mathbb{R}^{q})$ and the positive number $\eta$, there is a Borel map $T:\mathbb{R}^{q}\to\mathbb{R}^{q}$ whose image $T(\mathbb{R}^{q})$ is a finite set and

Fix such a $T$.

Step 4 (Cells). The set $F=T(\mathbb{R}^{q})$ is finite and nonempty (it contains $T(0_{\mathbb{R}^{q}})$), so by Finite Set and Number of Elements of a Set there are $M\in\mathbb{N}$ and a bijection $[M]\to F$, $i\mapsto z_{i}$; thus $z_{1},\dots,z_{M}$ are pairwise distinct and $F=\{z_{1},\dots,z_{M}\}$. For $i\in[M]$ put $s_{i}=\sum_{j=1}^{M}\lVert z_{i}-z_{j}\rVert$, and put $D=\sum_{i=1}^{M}s_{i}$. All summands being nonnegative, $0\le s_{i}$ by claim 5 of Properties of Finite Sums, and claim 6 there, applied twice, gives $\lVert z_{i}-z_{j}\rVert\le s_{i}\le D$ for all $i,j\in[M]$. For $i\in[M]$ put $A_{i}=T^{-1}(\{z_{i}\})$; since $\{z_{i}\}\in\mathcal{B}(\mathbb{R}^{q})$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §finite-sets and $T$ is Borel, $A_{i}\in\mathcal{B}(\mathbb{R}^{q})$. Put $c_{i}=\nu(A_{i})$; by monotonicity of measures (claim 2 of Basic Properties of a Measure) $0\le c_{i}\le\nu(\mathbb{R}^{q})=1$.

Step 5 (The threshold $N_{0}$). Put $m=\sum_{i=1}^{M}1$, a real number with $0\le m$ by claim 1 of Elementary Arithmetic in an Ordered Field and claim 5 of Properties of Finite Sums. Taking $i=j$ above gives $0\le\lVert z_{i}-z_{i}\rVert\le D$ (claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$), hence $0\le D$, so $0\le D^{2}$ and then $0\le D^{2}m$ by the second axiom of Ordered Field. Put $K=D^{2}m+1$. By claim 6 of Elementary Order Arithmetic in an Ordered Field, $0<1$, so $1\le K$ by the first axiom of Ordered Field and $0<K$ by claim 2 of Elementary Order Arithmetic in an Ordered Field; hence $K^{-1}$ exists and $0<K^{-1}$ by claim 7 there. Also $D^{2}m\le K$, since $0\le1$. Put $\beta=\eta^{2}K^{-1}$; by claim 5 of Elementary Order Arithmetic in an Ordered Field, $0<\eta^{2}$, $0<\beta$ and $0<\beta^{2}$, and $K\beta=\eta^{2}$. By claim 2 of The Archimedean Property of the Real Numbers, applied with $x=1$ and the positive number $\beta^{2}$, choose $N_{0}\in\mathbb{N}$ with $1<\iota(N_{0})\beta^{2}$.

We claim that $\iota(N)^{-1}<\beta^{2}$ for every $N\in\mathbb{N}$ with $N_{0}\le N$. Indeed, either $N=N_{0}$, or $N_{0}<N$ and then $\iota(N_{0})<\iota(N)$ by claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field; in both cases $\iota(N_{0})\le\iota(N)$, so $\iota(N_{0})\beta^{2}\le\iota(N)\beta^{2}$ by claim 5 of Elementary Arithmetic in an Ordered Field, and $1<\iota(N)\beta^{2}$ by claim 2 of Elementary Order Arithmetic in an Ordered Field. Multiplying by $\iota(N)^{-1}>0$ (claim 10 there) gives $\iota(N)^{-1}<\beta^{2}$.

Step 6 (Three auxiliary functions). From now on fix $N\in\mathbb{N}$ with $N_{0}\le N$; it remains to show $\phi_{N}(\nu)<\varepsilon$. By the first sentence of Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §quantisation, applied to $\nu$ and, for each $x\in\mathbb{R}^{qN}$, to $\mu^{N}_{x}\in\mathcal{P}(\mathbb{R}^{q})$, the measures $T_{\#}\nu$ and $T_{\#}\mu^{N}_{x}$ give measure $0$ to $\mathbb{R}^{q}\setminus F$. Hence The Wasserstein Distance between Two Probability Measures Carried by a Finite Set is Bounded by the Squared Diameter Times the Total Variation of the Masses §bound, applied with $n=q$, the points $z_{1},\dots,z_{M}$, the number $D$ of Step 4 and $\rho=T_{\#}\mu^{N}_{x}$, $\rho'=T_{\#}\nu$, shows $T_{\#}\nu\in\mathcal{P}_{2}(\mathbb{R}^{q})$, $T_{\#}\mu^{N}_{x}\in\mathcal{P}_{2}(\mathbb{R}^{q})$, and

the equality because $T_{\#}\rho(\{z_{i}\})=\rho(T^{-1}(\{z_{i}\}))=\rho(A_{i})$ for every $\rho\in\mathcal{P}(\mathbb{R}^{q})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward. By Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §pushforward with $p=q$ and $h=T$, $T_{\#}\mu^{N}_{x}=\mu^{N}_{T^{\oplus}(x)}$, where $T^{\oplus}:\mathbb{R}^{qN}\to\mathbb{R}^{qN}$ is the product map of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §product-map, which is Borel by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §product-map.

Define on $\mathbb{R}^{qN}$

and the real number $c=W_{2}(T_{\#}\nu,\nu)$, defined because $T_{\#}\nu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{q})$. The function $f_{N}\circ T^{\oplus}$ is Borel as a composition of Borel maps (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), so $u$ is Borel by claims 2 and 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. By Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §distance, applied with the measure $T_{\#}\nu\in\mathcal{P}_{2}(\mathbb{R}^{q})$ in place of $\nu$, the function $y\mapsto W_{2}(\mu^{N}_{y},T_{\#}\nu)$ on $\mathbb{R}^{qN}$ is Borel, and $g$ is its composition with $T^{\oplus}$, hence Borel. The functions $u$, $g$ and the constant $c$ (claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) are nonnegative random variables on $(\mathbb{R}^{qN},\mathcal{B}(\mathbb{R}^{qN}),P_{N})$, and by the display above, for every $x\in\mathbb{R}^{qN}$,

Step 7 (Pointwise domination). Let $x\in\mathbb{R}^{qN}$. By claim 3 of Properties of the Absolute Value in an Ordered Field, $f_{N}(x)-f_{N}(T^{\oplus}(x))\le u(x)$, so $f_{N}(x)\le u(x)+f_{N}(T^{\oplus}(x))$ by the first axiom of Ordered Field. Since $\mu^{N}_{T^{\oplus}(x)}\in\mathcal{P}_{2}(\mathbb{R}^{q})$ by Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §moment, the triangle inequality The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §triangle gives

Hence $0\le f_{N}(x)\le u(x)+g(x)+c$, and by (S) $f_{N}(x)^{2}\le(u(x)+g(x)+c)^{2}$.

Step 8 (The first function). Let $x\in\mathbb{R}^{qN}$. By Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §distance with $x'=T^{\oplus}(x)$, $u(x)\le W_{2}(\mu^{N}_{x},\mu^{N}_{T^{\oplus}(x)})$, so by (S) $u(x)^{2}\le W_{2}(\mu^{N}_{x},\mu^{N}_{T^{\oplus}(x)})^{2}$. By Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §lipschitz, $\iota(N)W_{2}(\mu^{N}_{x},\mu^{N}_{T^{\oplus}(x)})^{2}\le\lVert x-T^{\oplus}(x)\rVert^{2}$, and multiplying by $\iota(N)^{-1}\ge0$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $u(x)^{2}\le\iota(N)^{-1}\lVert x-T^{\oplus}(x)\rVert^{2}$. Let $g_{0}:\mathbb{R}^{q}\to\mathbb{R}^{q}$, $g_{0}(y)=y-T(y)$; its components are differences of components of the Borel maps $\mathrm{id}$ and $T$, so $g_{0}$ is Borel by 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 and claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and $\lVert g_{0}(y)\rVert^{2}=\lVert T(y)-y\rVert^{2}$. By Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear, $x=[\mathfrak{p}_{1}(x),\dots,\mathfrak{p}_{N}(x)]$, and with the definition Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §product-map of $T^{\oplus}(x)=[T(\mathfrak{p}_{1}(x)),\dots,T(\mathfrak{p}_{N}(x))]$ and $g_{0}^{\oplus}(x)=[g_{0}(\mathfrak{p}_{1}(x)),\dots,g_{0}(\mathfrak{p}_{N}(x))]$, the linearity of configurations in the same clause (with $t=-1$) gives $g_{0}^{\oplus}(x)=x-T^{\oplus}(x)$. Hence $u^{2}\le\iota(N)^{-1}\lVert g_{0}^{\oplus}\rVert^{2}$ pointwise, the right side being Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions and Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §product-map. By Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §product-maps with $P=P_{N}$, $p=q$ and $g=g_{0}$, together with $P_{N}^{[1]}=\nu$ from Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §marginal-of-tensor,

By claim 1 of Linearity and Monotonicity of the Lebesgue Integral (monotonicity and homogeneity) and Step 3,

Thus $u$ is square-integrable and $\lVert u\rVert_{2}\le\eta$ by (S).

Step 9 (The second function). For $i\in[M]$, by Variance of the Empirical Mass of a Borel Set under a Tensor Power §borel applied with $\rho=\nu$ and $B=A_{i}$ (so that its $c$ is our $c_{i}$), the map $x\mapsto\mu^{N}_{x}(A_{i})$ is Borel with values in $[0,1]$, so $h_{i}$ is Borel by claims 1, 2 and 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, with $0\le h_{i}\le1$ (claims 1 and 6 of Properties of the Absolute Value in an Ordered Field, as both $\mu^{N}_{x}(A_{i})$ and $c_{i}$ lie in $[0,1]$). Put $a_{i}=\int_{\mathbb{R}^{qN}}h_{i}\,dP_{N}$; by claim 1 of Linearity and Monotonicity of the Lebesgue Integral and (C), $0\le a_{i}\le1$, so $a_{i}$ is a nonnegative real number. By Variance of the Empirical Mass of a Borel Set under a Tensor Power §deviation, $a_{i}^{2}\le c_{i}\iota(N)^{-1}$; since $c_{i}\le1$ and $0\le\iota(N)^{-1}$, claim 5 of Elementary Arithmetic in an Ordered Field gives $c_{i}\iota(N)^{-1}\le\iota(N)^{-1}$, and Step 5 gives $\iota(N)^{-1}<\beta^{2}$; so $a_{i}^{2}<\beta^{2}$ by claim 2 of Elementary Order Arithmetic in an Ordered Field, and $a_{i}<\beta$ by (S). Hence $0\le\beta-a_{i}$ for every $i$ (claim 3 of Elementary Arithmetic in an Ordered Field); by claim 5 of Properties of Finite Sums, then its claims 2 and 3 (with $\lambda=-1$), $0\le\sum_{i=1}^{M}\beta-\sum_{i=1}^{M}a_{i}$, and $\sum_{i=1}^{M}\beta=\beta m$ by claim 3 there; so $\sum_{i=1}^{M}a_{i}\le\beta m$ by claim 3 of Elementary Arithmetic in an Ordered Field.

By Step 6 and claim 1 of Linearity and Monotonicity of the Lebesgue Integral (monotonicity, homogeneity, and additivity extended to the $M$ summands along the recursion of claim 1 of Properties of Finite Sums),

using claim 5 of Elementary Arithmetic in an Ordered Field twice (with the nonnegative multipliers $D^{2}$ and $\beta$, and $D^{2}m\le K$ from Step 5). Thus $g$ is square-integrable and $\lVert g\rVert_{2}\le\eta$ by (S).

Step 10 (The constant). By Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §pushforward with $\mu=\nu$, $S=T$ and the identity in place of its $T$, the measure $(T,\mathrm{id})_{\#}\nu$ is a coupling of $T_{\#}\nu$ and $\mathrm{id}_{\#}\nu=\nu$ (the latter because $\mathrm{id}^{-1}(B)=B$, Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward), with quadratic cost $\int_{\mathbb{R}^{q}}\lVert T(y)-y\rVert^{2}\,\nu(dy)\le\eta^{2}$ (Step 3). By The Quadratic Wasserstein Distance on Euclidean Space §distance, $c^{2}\le\eta^{2}$, so $c\le\eta$ by (S). By (C), $\mathbb{E}[c^{2}]=c^{2}$, so the constant $c$ is square-integrable and $\lVert c\rVert_{2}=c$ by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root.

Step 11 (Triangle inequality in mean square). By Square-Integrable Random Variables and the Mean-Square Inner Product, $u+g$ and $u+g+c$ are square-integrable, and claim 2 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm, applied twice on $(\mathbb{R}^{qN},\mathcal{B}(\mathbb{R}^{qN}),P_{N})$, gives

the last step by Steps 8, 9 and 10 and the first axiom of Ordered Field. By Step 7 and claim 1 of Linearity and Monotonicity of the Lebesgue Integral (monotonicity), $\phi_{N}(\nu)^{2}=\int_{\mathbb{R}^{qN}}f_{N}^{2}\,dP_{N}\le\mathbb{E}[(u+g+c)^{2}]=\lVert u+g+c\rVert_{2}^{2}$, so $\phi_{N}(\nu)\le\lVert u+g+c\rVert_{2}$ by (S). With Step 2 and claim 2 of Elementary Order Arithmetic in an Ordered Field, $\phi_{N}(\nu)<\varepsilon$.

Step 12 (Conclusion). Since $0\le\phi_{N}(\nu)$ (Step 1), $|\phi_{N}(\nu)-0|=\phi_{N}(\nu)$ by Absolute Value in an Ordered Field, so $|\phi_{N}(\nu)-0|<\varepsilon$ for every $N\in\mathbb{N}$ with $N_{0}\le N$. As $\varepsilon>0$ was arbitrary and $N_{0}$ was chosen in Step 5 depending only on $\varepsilon$ (through $\eta$, $T$, $D$, $M$ and $\beta$), the sequence $(\phi_{N}(\nu))_{N\in\mathbb{N}}$ converges to $0$ in the sense of Limit of a Sequence of Real Numbers. This proves clause 2.