Proof of Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals

Each result cited is universally quantified over the data in its own statement. For $n\in\mathbb{N}$ write $\mathfrak{p}^{(n)}_{k}$ ($k\in[n]$) for the block maps of $\mathbb{R}^{qn}$ given by Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks with $n$ in place of $N$, so that $\mathfrak{p}_{k}=\mathfrak{p}^{(N)}_{k}$. Each $\mathcal{B}(\mathbb{R}^{m})$ is a $\sigma$-algebra (Euclidean Space and Lebesgue Measure: Standing Notation §borel), hence closed under complements and countable unions by Sigma-Algebra and Measurable Space, and therefore under finite intersections. Finite sums in $[0,\infty]$ are the iterates given by Existence and Uniqueness of Iterates of a Binary Operation for the addition of $[0,\infty]$ fixed in Measure, Measure Space, and Probability Measure; by the uniqueness there, $\sum_{k=1}^{1}c_{k}=c_{1}$ and $\sum_{k=1}^{n+1}c_{k}=\sum_{k=1}^{n}c_{k}+c_{n+1}$, the sum up to $n$ being that of the restriction of the family to $[n]$, and these sums agree with the finite sums of real numbers when all summands are real. A natural number $n$ is read in $\mathbb{R}$ through the canonical map, which is positive and invertible by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, as fixed in The Real Numbers: Standing Notation and Background §numbers.

Step 1 (Block rectangles form a generating $\pi$-system). Let $\mathcal{C}$ be the family of the sets $\bigcap_{k=1}^{N}\mathfrak{p}_{k}^{-1}(B_{k})$ with $B_{1},\dots,B_{N}\in\mathcal{B}(\mathbb{R}^{q})$. Each $\mathfrak{p}_{k}$ is Borel by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear, so $\mathcal{C}\subseteq\mathcal{B}(\mathbb{R}^{qN})$. Taking every $B_{k}=\mathbb{R}^{q}$ shows $\mathbb{R}^{qN}\in\mathcal{C}$, and

with $B_{k}\cap B'_{k}\in\mathcal{B}(\mathbb{R}^{q})$; so $\mathcal{C}$ is a $\pi$-system in the sense of Dynkin's Pi-Lambda Theorem. By Generated Sigma-Algebra, $\sigma(\mathcal{C})\subseteq\mathcal{B}(\mathbb{R}^{qN})$. Conversely, let $R=A_{1}\times\dots\times A_{qN}$ with $A_{j}\in\mathcal{B}(\mathbb{R})$ be a Borel rectangle of $\mathbb{R}^{qN}$ in the sense of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$, points being read as tuples by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces. For $k\in[N]$ let $B_{k}=A_{b(k,1)}\times\dots\times A_{b(k,q)}$, the indices lying in $[qN]$ by Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §range. Then $B_{k}$ is a Borel rectangle of $\mathbb{R}^{q}$, hence belongs to the $\sigma$-algebra $\mathcal{B}_{q}$ generated by these rectangles (claim 1 of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$ and Generated Sigma-Algebra), which is $\mathcal{B}(\mathbb{R}^{q})$ by claim 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets. For $x\in\mathbb{R}^{qN}$, $x\in R$ means $x_{j}\in A_{j}$ for every $j\in[qN]$; since every $j\in[qN]$ equals $b(k,i)$ for some $k\in[N]$ and $i\in[q]$ (Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §bijection) and every such $b(k,i)$ lies in $[qN]$, this holds exactly when $x_{b(k,i)}\in A_{b(k,i)}$ for all $k\in[N]$ and $i\in[q]$, that is, by Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks, when $\mathfrak{p}_{k}(x)\in B_{k}$ for every $k\in[N]$. Thus $R\in\mathcal{C}$. Hence $\sigma(\mathcal{C})$ is a $\sigma$-algebra containing every Borel rectangle of $\mathbb{R}^{qN}$, so it contains the $\sigma$-algebra $\mathcal{B}_{qN}$ generated by them (claim 1 of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$ and Generated Sigma-Algebra), which is $\mathcal{B}(\mathbb{R}^{qN})$ by claim 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets. Therefore $\sigma(\mathcal{C})=\mathcal{B}(\mathbb{R}^{qN})$.

Step 2 (Uniqueness in claim 1). Let $P,P'\in\mathcal{P}(\mathbb{R}^{qN})$ both satisfy the display of claim 1. Then $P(C)=P'(C)$ for every $C\in\mathcal{C}$, and $P(\mathbb{R}^{qN})=P'(\mathbb{R}^{qN})=1<\infty$. By Step 1 and claim 1 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law, $P=P'$.

Step 3 (Existence in claim 1). Fix $\rho\in\mathcal{P}(\mathbb{R}^{q})$ and let $T$ be the set of those $n\in\mathbb{N}$ for which there is $P_{n}\in\mathcal{P}(\mathbb{R}^{qn})$ with $P_{n}\bigl(\bigcap_{k=1}^{n}(\mathfrak{p}^{(n)}_{k})^{-1}(B_{k})\bigr)=\prod_{k=1}^{n}\rho(B_{k})$ for all $B_{1},\dots,B_{n}\in\mathcal{B}(\mathbb{R}^{q})$.

$1\in T$: $q\cdot1=q$ by Natural Numbers, and $b(1,i)=(1-1)q+i=i$ for $i\in[q]$, since $1-1=0$, $0\cdot q=0$ and $0+i=i$ by the axioms of Field; so $\mathfrak{p}^{(1)}_{1}(x)=(x_{1},\dots,x_{q})=x$ for $x\in\mathbb{R}^{q}$ by Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks. As $[1]=\{1\}$ (claim 2 of Basic Properties of Initial Segments of the Natural Numbers), $\bigcap_{k=1}^{1}(\mathfrak{p}^{(1)}_{k})^{-1}(B_{k})=B_{1}$, and $\prod_{k=1}^{1}\rho(B_{k})=\rho(B_{1})$ by Finite Product Notation; so $P_{1}=\rho$ qualifies.

$n\in T$ implies $n+1\in T$: let $P_{n}$ be as in the definition of $T$. By Natural Numbers, $q(n+1)=qn+q$, so $\mathbb{R}^{q(n+1)}=\mathbb{R}^{qn+q}$ as in Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §concatenation. Let $Q=P_{n}\boxtimes\rho\in\mathcal{P}(\mathbb{R}^{qn+q})$ be the product measure of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product, with $qn$ and $q$ in place of $q$ and $p$, and write $\iota=\iota^{qn,q}$, $\mathrm{pr}_{1}=\mathrm{pr}^{qn,q}_{1}$, $\mathrm{pr}_{2}=\mathrm{pr}^{qn,q}_{2}$. Let $B_{1},\dots,B_{n+1}\in\mathcal{B}(\mathbb{R}^{q})$ and $A=\bigcap_{k=1}^{n}(\mathfrak{p}^{(n)}_{k})^{-1}(B_{k})$, which belongs to $\mathcal{B}(\mathbb{R}^{qn})$ by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear. For $z\in\mathbb{R}^{qn+q}$ put $u=\mathrm{pr}_{1}(z)$ and $v=\mathrm{pr}_{2}(z)$; then $z=\iota(u,v)$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, so $\mathfrak{p}^{(n+1)}_{k}(z)=\mathfrak{p}^{(n)}_{k}(u)$ for $k\in[n]$ and $\mathfrak{p}^{(n+1)}_{n+1}(z)=v$ by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §concatenation. Since $[n+1]=[n]\cup\{n+1\}$ (claim 3 of Basic Properties of Initial Segments of the Natural Numbers), $z\in\bigcap_{k=1}^{n+1}(\mathfrak{p}^{(n+1)}_{k})^{-1}(B_{k})$ holds exactly when $u\in A$ and $v\in B_{n+1}$; that is,

By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product its $Q$-measure is $P_{n}(A)\,\rho(B_{n+1})=\bigl(\prod_{k=1}^{n}\rho(B_{k})\bigr)\rho(B_{n+1})$, which is $\prod_{k=1}^{n+1}\rho(B_{k})$ by Finite Product Notation, as $2\le n+1$ by claims 4 and 6 of Properties of the Order on the Natural Numbers. So $P_{n+1}=Q$ qualifies and $n+1\in T$.

By Principle of Induction for the Natural Numbers, $T=\mathbb{N}$; in particular $N\in T$, which with Step 2 proves claim 1.

Step 4 (Finite sums of finite measures). Let $T'$ be the set of those $n\in\mathbb{N}$ with the following property: for every family $(\alpha_{k})_{k\in[n]}$ of measures on $(\mathbb{R}^{q},\mathcal{B}(\mathbb{R}^{q}))$ with every $\alpha_{k}(\mathbb{R}^{q})<\infty$, the function $S(B)=\sum_{k=1}^{n}\alpha_{k}(B)$ on $\mathcal{B}(\mathbb{R}^{q})$ (a sum of real numbers, as $\alpha_{k}(B)\le\alpha_{k}(\mathbb{R}^{q})$ by claim 2 of Basic Properties of a Measure) satisfies: (a) $S$ is a measure with $S(\mathbb{R}^{q})<\infty$; (b) $\int f\,dS=\sum_{k=1}^{n}\int f\,d\alpha_{k}$ in $[0,\infty]$ for every Borel $f:\mathbb{R}^{q}\to[0,\infty]$; (c) a Borel $f:\mathbb{R}^{q}\to\mathbb{R}$ is integrable with respect to $S$ exactly when it is integrable with respect to every $\alpha_{k}$, $k\in[n]$, and then $\int f\,dS=\sum_{k=1}^{n}\int f\,d\alpha_{k}$.

$1\in T'$: by claim 1 of Properties of Finite Sums, $S=\alpha_{1}$, and (a)--(c) are immediate.

$n\in T'$ implies $n+1\in T'$: let $(\alpha_{k})_{k\in[n+1]}$ be such a family, let $S'$ be the function formed from its restriction to $[n]\subseteq[n+1]$ (claim 3 of Basic Properties of Initial Segments of the Natural Numbers), which satisfies (a)--(c) because $n\in T'$, and let $S$ be formed from the whole family. By claim 1 of Properties of Finite Sums, $S(B)=S'(B)+\alpha_{n+1}(B)$. By Nonnegative Combinations of Two Finite Measures, and the Average of Two Couplings §combination with $E=\mathbb{R}^{q}$, $\alpha=S'$, $\beta=\alpha_{n+1}$ and $s=t=1$, $S$ is a measure with $S(\mathbb{R}^{q})<\infty$, and for Borel $f\ge0$, using $1\cdot c=c$ in $[0,\infty]$ (Measure Spaces and the Lebesgue Integral: Standing Notation §extended) and (b) for $S'$,

If a Borel $f:\mathbb{R}^{q}\to\mathbb{R}$ is integrable with respect to every $\alpha_{k}$, $k\in[n+1]$, then by (c) for $S'$ it is integrable with respect to $S'$ with $\int f\,dS'=\sum_{k=1}^{n}\int f\,d\alpha_{k}$, so by Nonnegative Combinations of Two Finite Measures, and the Average of Two Couplings §combination it is integrable with respect to $S$ and $\int f\,dS=\int f\,dS'+\int f\,d\alpha_{n+1}=\sum_{k=1}^{n+1}\int f\,d\alpha_{k}$. Conversely, if $f$ is integrable with respect to $S$, then $|f|$ is Borel (claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) and $\int|f|\,dS<\infty$ (Integrable Function and the Lebesgue Integral); as $\int|f|\,dS=\int|f|\,dS'+\int|f|\,d\alpha_{n+1}$ by Nonnegative Combinations of Two Finite Measures, and the Average of Two Couplings §combination and $a+\infty=\infty+a=\infty$ in $[0,\infty]$ (Measure, Measure Space, and Probability Measure), both $\int|f|\,dS'$ and $\int|f|\,d\alpha_{n+1}$ are finite, so $f$ is integrable with respect to $S'$ and to $\alpha_{n+1}$ by Integrable Function and the Lebesgue Integral, and with respect to every $\alpha_{k}$, $k\in[n]$, by (c) for $S'$. So $n+1\in T'$, and $T'=\mathbb{N}$ by Principle of Induction for the Natural Numbers.

Step 5 (Claim 2). Let $P\in\mathcal{P}(\mathbb{R}^{qN})$. For $k\in[N]$ let $\alpha_{k}=(\mathfrak{p}_{k})_{\#}P$, which by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward is a probability measure on $\mathbb{R}^{q}$ with $\alpha_{k}(B)=P(\mathfrak{p}_{k}^{-1}(B))$ and satisfies $\int g\,d\alpha_{k}=\int g\circ\mathfrak{p}_{k}\,dP$ for Borel $g:\mathbb{R}^{q}\to[0,\infty]$, a Borel $g:\mathbb{R}^{q}\to\mathbb{R}$ being integrable with respect to $\alpha_{k}$ exactly when $g\circ\mathfrak{p}_{k}$ is integrable with respect to $P$, with the same identity. Let $S$ be the function of Step 4 for this family (with $n=N$). By Nonnegative Combinations of Two Finite Measures, and the Average of Two Couplings §combination with $\alpha=\beta=S$, $s=N^{-1}$ and $t=0$, the function $B\mapsto N^{-1}S(B)+0\cdot S(B)=N^{-1}\sum_{k=1}^{N}P(\mathfrak{p}_{k}^{-1}(B))=A_{P}(B)$ is a measure. The set of $n\in\mathbb{N}$ with $\sum_{k=1}^{n}1=n$ contains $1$ and contains $n+1$ with $n$, by claim 1 of Properties of Finite Sums and claim 1 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, so it is $\mathbb{N}$ by Principle of Induction for the Natural Numbers; hence $A_{P}(\mathbb{R}^{q})=N^{-1}\sum_{k=1}^{N}P(\mathbb{R}^{qN})=N^{-1}N=1$, and $A_{P}\in\mathcal{P}(\mathbb{R}^{q})$.

For Borel $f:\mathbb{R}^{q}\to[0,\infty]$, the same application of Nonnegative Combinations of Two Finite Measures, and the Average of Two Couplings §combination, the conventions $0\cdot\infty=0$ and $c+0=c$ of Measure, Measure Space, and Probability Measure, Step 4(b) and the change of variables above give

Let $f:\mathbb{R}^{q}\to\mathbb{R}$ be Borel. If every $f\circ\mathfrak{p}_{k}$ is integrable with respect to $P$, then $f$ is integrable with respect to every $\alpha_{k}$, hence with respect to $S$ with $\int f\,dS=\sum_{k=1}^{N}\int f\,d\alpha_{k}$ by Step 4(c), hence, by the integrable case of Nonnegative Combinations of Two Finite Measures, and the Average of Two Couplings §combination (with $\alpha=\beta=S$, $s=N^{-1}$, $t=0$), with respect to $A_{P}$, with $\int f\,dA_{P}=N^{-1}\int f\,dS=\frac{1}{N}\sum_{k=1}^{N}\int f\circ\mathfrak{p}_{k}\,dP$. Conversely, if $f$ is integrable with respect to $A_{P}$, then $|f|$ is Borel (claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions) and $N^{-1}\int|f|\,dS=\int|f|\,dA_{P}<\infty$ (Integrable Function and the Lebesgue Integral); since $a\cdot\infty=\infty$ for positive $a$ (Measure Spaces and the Lebesgue Integral: Standing Notation §extended), $\int|f|\,dS<\infty$, so $f$ is integrable with respect to $S$ (Integrable Function and the Lebesgue Integral), hence with respect to every $\alpha_{k}$ by Step 4(c), and every $f\circ\mathfrak{p}_{k}$ is integrable with respect to $P$ by the change of variables above. This proves claim 2.