Proof of Continuity of the Coordinate Projections and of the Quadratic Cost Function, and Passage of a Cost Bound to a Weak Limit

Each result cited below is universally quantified over the data in its own statement, and is applied to the data named where it is used. Throughout, points of a Euclidean space are read as tuples by Euclidean Points as Tuples of Real Numbers, the difference of two of its points is formed componentwise by claim 1 of Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$ while its vector-space identities are those of Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, and two points with the same components are equal; the claims of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square are used for the Euclidean norm, claim 2 for $d_{E}(x,y)=\lVert x-y\rVert$, claim 5 for $\lVert\lambda x\rVert=|\lambda|\,\lVert x\rVert$ and claim 6 for the triangle inequality.

Claim 1. Write $\mathrm{pr}_{1}$ for $\mathrm{pr}^{q,p}_{1}$ and $\iota$ for $\iota^{q,p}$. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i) every $z\in\mathbb{R}^{q+p}$ satisfies $z=\iota(\mathrm{pr}_{1}(z),\mathrm{pr}^{q,p}_{2}(z))$, and by Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space the $i$th component of $\iota(x,y)$ is $x_{i}$ for $i$ among the first $q$ indices; hence

$$(\mathrm{pr}_{1}(z))_{i}=z_{i}\qquad\text{for each of the first }q\text{ indices }i .$$

Let $z,w\in\mathbb{R}^{q+p}$. For each such $i$,

$$\bigl(\mathrm{pr}_{1}(z)-\mathrm{pr}_{1}(w)\bigr)_{i}=(\mathrm{pr}_{1}(z))_{i}-(\mathrm{pr}_{1}(w))_{i}=z_{i}-w_{i}=(z-w)_{i}=\bigl(\mathrm{pr}_{1}(z-w)\bigr)_{i},$$

the first and third equalities by claim 1 of Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$ and the last by the displayed identity applied to $z-w$. Hence $\mathrm{pr}_{1}(z)-\mathrm{pr}_{1}(w)=\mathrm{pr}_{1}(z-w)$, and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i) gives

$$d_{E}(\mathrm{pr}_{1}(z),\mathrm{pr}_{1}(w))=\lVert\mathrm{pr}_{1}(z-w)\rVert\le\lVert z-w\rVert=d_{E}(z,w).$$

So $\mathrm{pr}_{1}$ is Lipschitz with constant $1$ in the sense of Lipschitz Map Between Metric Spaces; it is uniformly continuous by A Lipschitz Map is Uniformly Continuous and Continuous and continuous by A Uniformly Continuous Map Between Metric Spaces Is Continuous. The argument for $\mathrm{pr}^{q,p}_{2}$ is the same, with the components of $\iota(x,y)$ at the last $p$ indices, which are those of $y$ by Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space.

Claim 2. Let $D:\mathbb{R}^{m+m}\to\mathbb{R}^{m}$ be the map with $D(z)=\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)$. For $z,w\in\mathbb{R}^{m+m}$ the vector identities of Euclidean Space $\mathbb{R}^n$ is a Real Vector Space give

$$D(z)-D(w)=\bigl(\mathrm{pr}_{1}(z)-\mathrm{pr}_{1}(w)\bigr)-\bigl(\mathrm{pr}_{2}(z)-\mathrm{pr}_{2}(w)\bigr),$$

so by the triangle inequality, by $\lVert-x\rVert=\lVert x\rVert$ and by claim 1,

$$\lVert D(z)-D(w)\rVert\le\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{1}(w)\rVert+\lVert\mathrm{pr}_{2}(z)-\mathrm{pr}_{2}(w)\rVert\le\lVert z-w\rVert+\lVert z-w\rVert=2\lVert z-w\rVert .$$

Thus $D$ is Lipschitz with constant $2$, hence continuous on $\mathbb{R}^{m+m}$ by A Lipschitz Map is Uniformly Continuous and Continuous and A Uniformly Continuous Map Between Metric Spaces Is Continuous.

Let $N:\mathbb{R}^{m}\to\mathbb{R}$ be the map with $N(x)=\lVert x\rVert$, which is continuous on $\mathbb{R}^{m}$ by The Radial Compactification of Euclidean Space: Norm Continuity, a Homeomorphism onto the Open Unit Ball, Radial Cutoffs, and Extension of Test Functions (i). By claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map, applied with the continuous map $D$ and the continuous real-valued map $N$, the composition $N\circ D$ is continuous on $\mathbb{R}^{m+m}$. Since $\lVert x\rVert^{2}=\lVert x\rVert\,\lVert x\rVert$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), the map $\varphi$ is the pointwise product $(N\circ D)(N\circ D)$, which is continuous on $\mathbb{R}^{m+m}$ by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space.

Claim 3. Let $\iota_{\mathbb{N}}$ be the embedding of The Canonical Map from the Natural Numbers to a Field of the natural numbers in $\mathbb{R}$, and for $k\in\mathbb{N}$ put $t_{k}=\iota_{\mathbb{N}}(k)$ and let $g_{k}:\mathbb{R}^{m+m}\to\mathbb{R}$ be the map with

$$g_{k}(z)=\min\{\varphi(z),t_{k}\}.$$

The $g_{k}$ are bounded continuous test functions. By claim 2 the map $\varphi$ is continuous, and constants are continuous by claim 1 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, so $g_{k}$ is continuous on $\mathbb{R}^{m+m}$ by claim 4 of Continuity of the Absolute Value, Maximum and Minimum of Real-Valued Functions on a Metric Space. By Minimum of Two Elements of a Totally Ordered Set and Elementary Properties of the Minimum of Two Elements, $g_{k}(z)\le t_{k}$ and $g_{k}(z)$ is one of $\varphi(z)$ and $t_{k}$, both nonnegative, so $0\le g_{k}(z)\le t_{k}$; hence $g_{k}$ is bounded, and it is Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space.

The $g_{k}$ increase to $\varphi$. For every $z$, $g_{k}(z)\le\varphi(z)$ by Elementary Properties of the Minimum of Two Elements; and $t_{k}\le t_{k+1}$ by claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, so $g_{k}(z)\le g_{k+1}(z)$ by claim 3 of Elementary Properties of the Minimum of Two Elements, since $g_{k}(z)\le\varphi(z)$ and $g_{k}(z)\le t_{k}\le t_{k+1}$. Moreover $\varphi(z)$ is the least upper bound of $\{g_{k}(z):k\in\mathbb{N}\}$: it is an upper bound, and by The Archimedean Property of the Real Numbers there is $k\in\mathbb{N}$ with $\varphi(z)<t_{k}$, for which $g_{k}(z)=\varphi(z)$, so no smaller number is an upper bound.

A bound for each $k$. Fix $k\in\mathbb{N}$. Read as maps into $[0,\infty]$, which by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii) does not change measurability and returns the same integral for a nonnegative real-valued map, the maps $g_{k}$ and $\varphi$ satisfy $g_{k}\le\varphi$ pointwise, so claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion gives $\int_{\mathbb{R}^{m+m}}g_{k}\,d\pi_{n}\le J(\pi_{n})$ for every $n\in\mathbb{N}$.

Let $\varepsilon\in\mathbb{R}$ be positive and let $N\in\mathbb{N}$ be as in the hypothesis for this $\varepsilon$. Put $a_{n}=\int_{\mathbb{R}^{m+m}}g_{k}\,d\pi_{n}$ and $a=\int_{\mathbb{R}^{m+m}}g_{k}\,d\pi$; thus $a_{n}\le c+\varepsilon$ for every $n$ with $N\le n$. Since $g_{k}$ is bounded and continuous and $(\pi_{n})_{n\in\mathbb{N}}$ converges weakly to $\pi$, Weak Convergence of Finite Borel Measures on a Metric Space gives that the sequence $(a_{n})_{n\in\mathbb{N}}$ converges to $a$.

Suppose, for contradiction, that $c+\varepsilon<a$, and put $\eta=a-(c+\varepsilon)$, a positive real number. By the definition of the limit there is $N'\in\mathbb{N}$ with $|a_{n}-a|<\eta$ for every $n$ with $N'\le n$. Put $n=N+N'$; by claim 6 of Properties of the Order on the Natural Numbers we have $N<N+N'$ and $N'<N'+N$, and $N'+N=N+N'$ by the commutativity of addition in $\mathbb{N}$, so $N\le n$ and $N'\le n$. Then $a-a_{n}\le|a-a_{n}|=|a_{n}-a|<\eta$ by claims 3 and 2 of Properties of the Absolute Value in an Ordered Field, so $c+\varepsilon=a-\eta<a_{n}$, contradicting $a_{n}\le c+\varepsilon$. Hence $a\le c+\varepsilon$.

As $\varepsilon$ was an arbitrary positive real number, claim 1 of Comparison of Real Numbers with Arbitrary Positive Slack gives

$$\int_{\mathbb{R}^{m+m}}g_{k}\,d\pi\le c\qquad\text{for every }k\in\mathbb{N}.$$

Passage to the limit in $k$. The maps $g_{k}$, read as maps into $[0,\infty]$, form a nondecreasing sequence of nonnegative measurable maps whose pointwise least upper bound is $\varphi$. By Monotone Convergence Theorem, applied on the measure space $(\mathbb{R}^{m+m},\mathcal{B}(\mathbb{R}^{m+m}),\pi)$,

$$J(\pi)=\int_{\mathbb{R}^{m+m}}\varphi\,d\pi=\sup_{k\in\mathbb{N}}\int_{\mathbb{R}^{m+m}}g_{k}\,d\pi .$$

The real number $c$ is an upper bound of the set of which the right-hand side is the least upper bound, and a least upper bound is at most every upper bound, so $J(\pi)\le c$; in particular $J(\pi)$ is not $\infty$, hence is a real number.