Proof of The Torus Displacement Pairing: Linearity, the Cost Bound, and Vanishing of a Field with First-Order Small Pairings

Each result cited is universally quantified over the data in its own statement, and is applied here to the data named in the statement above; results stated for a Euclidean dimension $n$, or for a natural number $d$ with $1\le d$, are applied with that number equal to the dimension $d$ fixed in Optimal Transport on the Flat Torus: Standing Notation §conventions.

Preliminaries. Measures in $\mathcal{P}(\mathbb{T}^{d})$ belong to $\mathcal{P}(\mathbb{R}^{d})$ by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §measures. For $\nu\in\mathcal{P}(\mathbb{T}^{d})$ and $\gamma\in\Pi(\mu,\nu)$, the measure $\gamma$ belongs to $\mathcal{P}(\mathbb{R}^{d+d})$ and satisfies $(\mathrm{pr}_{1})_{\#}\gamma=\mu$ and $(\mathrm{pr}_{2})_{\#}\gamma=\nu$ by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling; the projections are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections. By Optimal Transport on the Flat Torus: Standing Notation §fields, $L^{2}(\mu;\mathbb{R}^{d})$ is the space of classes of square-integrable random vectors on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$: a representative of $\eta\in L^{2}(\mu;\mathbb{R}^{d})$, again written $\eta$, is a Borel map $\mathbb{R}^{d}\to\mathbb{R}^{d}$ (Random Vector and Its Law §vector) with $\int_{\mathbb{R}^{d}}\lVert\eta\rVert^{2}\,d\mu<\infty$, and $\lVert\eta\rVert_{\mu}=\sqrt{\int_{\mathbb{R}^{d}}\lVert\eta\rVert^{2}\,d\mu}$ by The Space of Square-Integrable Random Vectors §inner-product. By The Space of Square-Integrable Random Vectors §classes, if $\eta,\eta'$ are represented by Borel maps and $a,b$ are real, then $a\eta+b\eta'$ is represented by the pointwise combination $x\mapsto a\,\eta(x)+b\,\eta'(x)$. For such a representative write $g_{\eta}(w)=\eta(\mathrm{pr}_{1}(w))\cdot\varpi(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w))$ for $w\in\mathbb{R}^{d+d}$; by the preamble of The Torus Displacement Pairing of a Vector Field Along a Coupling this function is Borel and $\gamma$-integrable for every coupling $\gamma$ as above, $\mathcal{J}_{\mathbb{T}}(\eta,\gamma)=\int g_{\eta}\,d\gamma$ by The Torus Displacement Pairing of a Vector Field Along a Coupling §pairing, and the value does not depend on the representative. Finally $d_{\mathbb{T}}(x,y)=\lVert\varpi(y-x)\rVert$ for all $x,y\in\mathbb{R}^{d}$ by The Wrapped Displacement and the Flat Torus Distance §distance, and by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures a nonnegative Borel function with values in $\mathbb{R}$ is integrated as a map into $[0,\infty]$, so its integral as an integrable function, when it is integrable, is that integral.

Claim 1. Let $\nu$, $\gamma$, $\eta$, $\eta'$, $a$, $b$ be as in claim 1, with Borel representatives $\eta,\eta'$. By the Preliminaries, $a\eta+b\eta'$ is represented by $x\mapsto a\,\eta(x)+b\,\eta'(x)$, so $\mathcal{J}_{\mathbb{T}}(a\eta+b\eta',\gamma)$ is the integral of the function whose value at $w$ is $\bigl(a\,\eta(\mathrm{pr}_{1}(w))+b\,\eta'(\mathrm{pr}_{1}(w))\bigr)\cdot\varpi(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w))$. By claims 2 and 4 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$ this value equals $a\,g_{\eta}(w)+b\,g_{\eta'}(w)$ for every $w$. Both $g_{\eta}$ and $g_{\eta'}$ are $\gamma$-integrable, so claim 2 of Linearity and Monotonicity of the Lebesgue Integral gives

$$\mathcal{J}_{\mathbb{T}}(a\eta+b\eta',\gamma)=a\int g_{\eta}\,d\gamma+b\int g_{\eta'}\,d\gamma=a\,\mathcal{J}_{\mathbb{T}}(\eta,\gamma)+b\,\mathcal{J}_{\mathbb{T}}(\eta',\gamma).$$

Claim 2. Let $\nu$, $\gamma$ and $\eta$ be as in claim 2, with a Borel representative $\eta$. The triple $(\mathbb{R}^{d+d},\mathcal{B}(\mathbb{R}^{d+d}),\gamma)$ is a probability space, since $\gamma(\mathbb{R}^{d+d})=1$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. Define $X,Y:\mathbb{R}^{d+d}\to\mathbb{R}$ by $X(w)=\lVert\eta(\mathrm{pr}_{1}(w))\rVert$ and $Y(w)=d_{\mathbb{T}}(\mathrm{pr}_{1}(w),\mathrm{pr}_{2}(w))$. The composite $\eta\circ\mathrm{pr}_{1}$ is Borel as a composition of Borel maps (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), and the norm $x\mapsto\lVert x\rVert$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, so $X$ is Borel; $Y$ is Borel by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz. Thus $X$ and $Y$ are nonnegative random variables on this probability space. The function $\lVert\eta\rVert^{2}$ on $\mathbb{R}^{d}$ is nonnegative and 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 change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, applied to $\mathrm{pr}_{1}$ with $(\mathrm{pr}_{1})_{\#}\gamma=\mu$, gives

$$\int_{\mathbb{R}^{d+d}}X^{2}\,d\gamma=\int_{\mathbb{R}^{d}}\lVert\eta\rVert^{2}\,d\mu=\lVert\eta\rVert_{\mu}^{2}<\infty,$$

while $\int Y^{2}\,d\gamma=I_{\mathbb{T}}(\gamma)\le d/4$ by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §cost. Hence $X$ and $Y$ are square-integrable, with mean-square norms $\lVert X\rVert_{2}=\lVert\eta\rVert_{\mu}$ and $\lVert Y\rVert_{2}=\sqrt{I_{\mathbb{T}}(\gamma)}$, and by the same definition the product $XY$ is integrable with respect to $\gamma$.

For every $w$, the Cauchy--Schwarz inequality Cauchy-Schwarz Inequality for the Euclidean Dot Product and the Preliminaries give

$$|g_{\eta}(w)|\le\lVert\eta(\mathrm{pr}_{1}(w))\rVert\,\lVert\varpi(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w))\rVert=X(w)\,Y(w).$$

The functions $|g_{\eta}|$ and $XY$ are nonnegative and Borel by claims 4 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. Therefore

$$|\mathcal{J}_{\mathbb{T}}(\eta,\gamma)|=\Bigl|\int g_{\eta}\,d\gamma\Bigr|\le\int|g_{\eta}|\,d\gamma\le\int XY\,d\gamma=\mathbb{E}[XY]\le\bigl|\mathbb{E}[XY]\bigr|\le\lVert X\rVert_{2}\,\lVert Y\rVert_{2}=\lVert\eta\rVert_{\mu}\sqrt{I_{\mathbb{T}}(\gamma)} ,$$

where the first inequality is claim 2 of Linearity and Monotonicity of the Lebesgue Integral, the second is the monotonicity in claim 1 of that theorem, the equality with the expectation holds by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §probability-space and the Preliminaries, the next inequality is claim 3 of Properties of the Absolute Value in an Ordered Field, and the last inequality is claim 1 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm.

Claim 3. Let $\eta$ be as in claim 3, fix a Borel representative, again written $\eta$, and let $0_{L^{2}}$ denote the zero element of $L^{2}(\mu;\mathbb{R}^{d})$. A natural number $M$ is read as a real number through the canonical map; so read, it is positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. Write $\tfrac12=2^{-1}$, which exists and is positive by claim 8 of Elementary Order Arithmetic in an Ordered Field.

Step 3a: truncations. Let $\phi(x)=\lVert\eta(x)\rVert$; as in claim 2, $\phi:\mathbb{R}^{d}\to\mathbb{R}$ is nonnegative and Borel. Fix $M\in\mathbb{N}$. By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §markov, applied in the measure space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$ to $\phi$, read as a measurable map into $[0,\infty]$ as in the Preliminaries, and to the positive real $M$, the set $\{x:M\le\phi(x)\}$ is Borel; so is its complement

$$B_{M}=\{x\in\mathbb{R}^{d}:\phi(x)<M\},$$

the order of $\mathbb{R}$ being total. Define $h_{M}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ by $h_{M}(x)=\mathbf{1}_{B_{M}}(x)\,\eta(x)$. Its components $\mathbf{1}_{B_{M}}\eta_{i}$ are Borel by claims 1 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, the components $\eta_{i}$ of the Borel map $\eta$ being Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps; hence $h_{M}$ is Borel by the same clause. If $x\in B_{M}$, then $h_{M}(x)=\eta(x)$ and $\lVert h_{M}(x)\rVert=\phi(x)<M$; if $x\notin B_{M}$, then $h_{M}(x)=0\,\eta(x)$ is the origin and $\lVert h_{M}(x)\rVert=0<M$ by claim 3 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$. So

$$\lVert h_{M}(x)\rVert<M\qquad(x\in\mathbb{R}^{d}). \tag{1}$$

Put $u_{M}(x)=\mathbf{1}_{B_{M}}(x)\,\lVert\eta(x)\rVert^{2}$. In both cases above, by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and claim 5 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$ (the dot product with the origin being $0$ by homogeneity with the scalar $0$),

$$\eta(x)\cdot h_{M}(x)=\lVert h_{M}(x)\rVert^{2}=u_{M}(x)\qquad(x\in\mathbb{R}^{d}). \tag{2}$$

The function $u_{M}$ is nonnegative and Borel (claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions) and satisfies $u_{M}\le\lVert\eta\rVert^{2}$ pointwise, so, by the monotonicity in claim 1 of Linearity and Monotonicity of the Lebesgue Integral,

$$A_{M}=\int_{\mathbb{R}^{d}}u_{M}\,d\mu\quad\text{satisfies}\quad0\le A_{M}\le\lVert\eta\rVert_{\mu}^{2}<\infty ;$$

in particular $u_{M}$ is $\mu$-integrable.

Step 3b: displacement couplings. Keep $M$ fixed and let $t$ be a real number with $0<t$ and $tM<\tfrac12$. Let $\mathrm{id}$ be the identity map of $\mathbb{R}^{d}$, which is Borel, as recorded in the preamble 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, and define $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ by $S(x)=\pi\bigl(x+t\,h_{M}(x)\bigr)$. The map $x\mapsto x+t\,h_{M}(x)$ is Borel, its $i$th component being the sum of the $i$th component of $\mathrm{id}$ and $t$ times the $i$th component of $h_{M}$ (claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps); $\pi$ is Borel by The Half-Open Unit Cell Tiles Euclidean Space §wrap; so $S$ is Borel as a composition (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps).

(i) Put $\nu_{t}=S_{\#}\mu$, which lies in $\mathcal{P}(\mathbb{R}^{d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward. Since $S(x)\in Q$ for every $x$ by The Half-Open Unit Cell Tiles Euclidean Space §wrap, and $Q$ is Borel by The Half-Open Unit Cell Tiles Euclidean Space §cell, one has $\nu_{t}(Q)=\mu(S^{-1}(Q))=\mu(\mathbb{R}^{d})=1$; so $\nu_{t}\in\mathcal{P}(\mathbb{T}^{d})$ by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §measures.

(ii) Put $\gamma_{t}=(\mathrm{id},S)_{\#}\mu$. 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, applied to the Borel maps $\mathrm{id}$ and $S$, $\gamma_{t}\in\Pi(\mathrm{id}_{\#}\mu,S_{\#}\mu)=\Pi(\mu,\nu_{t})$, since $\mathrm{id}_{\#}\mu(B)=\mu(\mathrm{id}^{-1}(B))=\mu(B)$ for every Borel $B$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward). The pairing $(\mathrm{id},S)$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, and $\mathrm{pr}_{1}((\mathrm{id},S)(x))=x$, $\mathrm{pr}_{2}((\mathrm{id},S)(x))=S(x)$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections.

(iii) Fix $x\in\mathbb{R}^{d}$. By The Half-Open Unit Cell Tiles Euclidean Space §wrap, $S(x)=x+t\,h_{M}(x)-m$ for an integer vector $m\in\mathbb{Z}^{d}$, so $S(x)-x=t\,h_{M}(x)+k$ with $k=-m$, whose coordinates $-m_{i}$ are integers, so that $k\in\mathbb{Z}^{d}$ (Lattice-Periodic Functions and the Periodic Function Classes §lattice). By The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range, $\varpi(S(x)-x)=\varpi(t\,h_{M}(x))$. Let $i\in[d]$ and let $s_{i}=t\,h_{M}(x)_{i}$ be the $i$th component of $t\,h_{M}(x)$ (Scalar Multiple of a Point of $\mathbb{R}^n$). By claims 4 and 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, with $|t|=t$ as $0\le t$ (Absolute Value in an Ordered Field), and by (1) and claim 5 of Elementary Arithmetic in an Ordered Field,

$$|s_{i}|\le\lVert t\,h_{M}(x)\rVert=t\,\lVert h_{M}(x)\rVert\le tM<\tfrac12 ,$$

so $|s_{i}|<\tfrac12$ by claim 2 of Elementary Order Arithmetic in an Ordered Field, that is $-\tfrac12<s_{i}<\tfrac12$ by claim 9 of Properties of the Absolute Value in an Ordered Field. Adding $\tfrac12$ (claim 1 of Elementary Order Arithmetic in an Ordered Field) and using $-\tfrac12+\tfrac12=0$ and $\tfrac12+\tfrac12=1$ (claim 8 there, with the positive element $1$) gives $0<s_{i}+\tfrac12<1$. Thus the integer $0$ satisfies $0\le s_{i}+\tfrac12<0+1$, so $\lfloor s_{i}+\tfrac12\rfloor=0$ by the uniqueness in Existence and Uniqueness of the Integer Part of a Real Number, and $\varpi(t\,h_{M}(x))_{i}=s_{i}-0=s_{i}$ by The Wrapped Displacement and the Flat Torus Distance §displacement. By claim 1 of Euclidean Points as Tuples of Real Numbers,

$$\varpi\bigl(S(x)-x\bigr)=t\,h_{M}(x)\qquad(x\in\mathbb{R}^{d}). \tag{3}$$

(iv) By the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, applied to the Borel map $(\mathrm{id},S)$ and to the $\gamma_{t}$-integrable function $g_{\eta}$ of the Preliminaries, and by (ii), (3), claim 5 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$ and (2),

$$\mathcal{J}_{\mathbb{T}}(\eta,\gamma_{t})=\int_{\mathbb{R}^{d}}\eta(x)\cdot\varpi\bigl(S(x)-x\bigr)\,\mu(dx)=\int_{\mathbb{R}^{d}}t\,\bigl(\eta(x)\cdot h_{M}(x)\bigr)\,\mu(dx)=\int_{\mathbb{R}^{d}}t\,u_{M}\,d\mu=t\,A_{M},$$

the last equality by claim 1 of Linearity and Monotonicity of the Lebesgue Integral with the nonnegative scalar $t$ and the last clause of the Preliminaries. Likewise, the same formula applied to the nonnegative Borel integrand of Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §cost, together with (ii), the Preliminaries, (3), claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and (2), gives

$$I_{\mathbb{T}}(\gamma_{t})=\int_{\mathbb{R}^{d}}d_{\mathbb{T}}\bigl(x,S(x)\bigr)^{2}\,\mu(dx)=\int_{\mathbb{R}^{d}}\lVert t\,h_{M}(x)\rVert^{2}\,\mu(dx)=\int_{\mathbb{R}^{d}}t^{2}\,u_{M}\,d\mu=t^{2}A_{M},$$

again by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, now with the nonnegative scalar $t^{2}$.

Step 3c: $A_{M}=0$ for every $M\in\mathbb{N}$. Fix $M$ and suppose, for a contradiction, that $0<A_{M}$. Let $s=\sqrt{A_{M}}$, the nonnegative square root; $s\ne0$, since otherwise $A_{M}=s^{2}=0$, so $0<s$. Let $\varepsilon$ be an arbitrary positive real number, and, in this order, let $\theta$ be a positive real supplied by the hypothesis of claim 3 for this $\varepsilon$; then put

$$t_{1}=M^{-1}\cdot\tfrac12\cdot\tfrac12,\qquad t_{2}=\theta\,s^{-1}\cdot\tfrac12 .$$

Both are positive by claims 7, 5 and 8 of Elementary Order Arithmetic in an Ordered Field, and $t_{1}M=\tfrac12\cdot\tfrac12<\tfrac12$ and $t_{2}s=\theta\cdot\tfrac12<\theta$ by claim 8 there (applied to the positive elements $\tfrac12$ and $\theta$). Let $t$ be the lesser of $t_{1}$ and $t_{2}$, supplied by claim 9 there; $t$ equals one of them, so $0<t$, and $t\le t_{1}$, $t\le t_{2}$. By claim 5 of Elementary Arithmetic in an Ordered Field with the nonnegative multipliers $M$ and $s$, and claim 2 of Elementary Order Arithmetic in an Ordered Field, $tM\le t_{1}M<\tfrac12$ and $ts\le t_{2}s<\theta$. So Step 3b applies to $t$, and $I_{\mathbb{T}}(\gamma_{t})=t^{2}A_{M}=(ts)^{2}<\theta^{2}$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, the numbers $ts$ and $\theta$ being nonnegative. The hypothesis of claim 3, applied to $\nu_{t}\in\mathcal{P}(\mathbb{T}^{d})$ and $\gamma_{t}\in\Pi(\mu,\nu_{t})$, yields $|\mathcal{J}_{\mathbb{T}}(\eta,\gamma_{t})|\le\varepsilon\sqrt{I_{\mathbb{T}}(\gamma_{t})}$. Here $\sqrt{I_{\mathbb{T}}(\gamma_{t})}=ts$ by Existence and Uniqueness of the Nonnegative Square Root, $ts$ being nonnegative with square $I_{\mathbb{T}}(\gamma_{t})$, and $|\mathcal{J}_{\mathbb{T}}(\eta,\gamma_{t})|=|tA_{M}|=tA_{M}=t\,s^{2}$ by Step 3b and Absolute Value in an Ordered Field, as $0<tA_{M}$ by claim 5 of Elementary Order Arithmetic in an Ordered Field. Hence $t\,s\cdot s\le\varepsilon\cdot ts$. The number $ts$ is positive by claim 5 there, so its inverse exists and is positive by claim 7 there, and multiplying by it (claim 5 of Elementary Arithmetic in an Ordered Field) gives $s\le\varepsilon$. As $\varepsilon$ was an arbitrary positive real and $0\le s$, Comparison of Real Numbers with Arbitrary Positive Slack §vanishing gives $s=0$, contradicting $0<s$. Hence $A_{M}\le0$, the order being total, and $A_{M}=0$ since $0\le A_{M}$.

Step 3d: conclusion. For every $M\in\mathbb{N}$, the nonnegative Borel function $u_{M}$ has $\int u_{M}\,d\mu=A_{M}=0$, so $u_{M}(x)=0$ for $\mu$-almost every $x$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §vanishing. By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-union, the set $N$ of points $x$ at which $u_{M}(x)\ne0$ for at least one $M\in\mathbb{N}$ is $\mu$-null. Let $x\in\mathbb{R}^{d}\setminus N$. By claim 1 of The Archimedean Property of the Real Numbers there is $M\in\mathbb{N}$ with $\phi(x)<M$, that is, $x\in B_{M}$; then $\lVert\eta(x)\rVert^{2}=u_{M}(x)=0$. Thus the nonnegative Borel function $\lVert\eta\rVert^{2}$ vanishes off the null set $N$, and $\int_{\mathbb{R}^{d}}\lVert\eta\rVert^{2}\,d\mu=0$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-integral. By the Preliminaries, $\lVert\eta\rVert_{\mu}$ is the nonnegative square root of $0$, which is $0$ by Existence and Uniqueness of the Nonnegative Square Root. Since $L^{2}(\mu;\mathbb{R}^{d})$ is a real inner product space with norm $\lVert\cdot\rVert_{\mu}$ (The Space of Square-Integrable Random Vectors §inner-product), Elementary Identities in a Real Inner Product Space §vanishing gives $\eta=0_{L^{2}}$.