Infinite Product Measures and Kolmogorov Extension

Put $H=F\cup G$ and let $p_F,p_G$ be the coordinate projections from ${\prod }_H{E}_i$. Then $C:=p_F^{-1}(A)\cap p_G^{-1}(B)$ is measurable in ${⨂}_H{\mathcal{E}}_i$.

Directly from restriction of coordinates, ${\pi }_F^{-1}(A)\cap {\pi }_G^{-1}(B)={\pi }_H^{-1}(C)$, a finite-coordinate cylinder. The family is nonempty because it contains $E$, so it is a pi-system.

The empty-support cylinders give $\mathrm{\varnothing },E\in {\mathcal{A}}_I$. Also $E\setminus {\pi }_F^{-1}(A)={\pi }_F^{-1}(({\prod }_F{E}_i)\setminus A)\in {\mathcal{A}}_I$.

For $H=F\cup G$, pull both bases to ${\prod }_H{E}_i$; their union is measurable and its inverse image is ${\pi }_F^{-1}(A)\cup {\pi }_G^{-1}(B)$. Thus ${\mathcal{A}}_I$ has the three algebra operations.

Fix $x^{\ast }\in E$. If ${\pi }_F^{-1}(A)={\pi }_G^{-1}(B)$, pull both sets to $H=F\cup G$. Their lifted bases are equal: otherwise a point of their symmetric difference, combined with the coordinates of $x^{\ast }$ outside $H$, would distinguish the cylinders. Hence consistency gives ${\mu }_F(A)={\mu }_H(p_{H,F}^{-1}A)={\mu }_H(p_{H,G}^{-1}B)={\mu }_G(B)$.

For disjoint cylinders $C_r={\pi }_{F_r}^{-1}(A_r)$ with $r<m$, pull all bases to $H={\bigcup }_{r<m}{F}_r$. They are disjoint measurable sets, so finite additivity of ${\mu }_H$ yields ${\mu }_0({\bigcup }_r{C}_r)={\sum }_r{\mu }_0(C_r)$. The union is in the cylinder algebra by [F2].

Every coordinate space is nonempty because it carries a probability measure. By [F4], the coordinate product is therefore nonempty, so the finitely additive cylinder law is well-defined. Let $C_n\downarrow \mathrm{\varnothing }$ be cylinders and suppose that ${\mu }_0(C_n)\ge \eta >0$ for every $n$. By [F5], enumerate the countable union of their finite supports. After enlarging supports, take $C_n$ to be determined by the first $k_n$ active coordinates, with $(k_n)$ nondecreasing. Recursively regarding each finite product as a two-factor product, [F2] expresses each cylinder mass as the integral of its measurable next-coordinate section-mass function.

The finite-stage section argument recursively maintains the following invariant after $m$ coordinates have been chosen: every remaining $C_n$-section has tail-cylinder mass at least $\eta \mathrm{/}{2}^m$. For a prefix with this invariant, let $D_n$ be the measurable set of possible next coordinates whose further section has mass at least $\eta \mathrm{/}{2}^{m+1}$. The $D_n$ decrease with $n$. The section formula and the bound by $1$ give the next-coordinate measure of $D_n$ at least $\eta \mathrm{/}{2}^{m+1}$; [F3] therefore makes ${\bigcap }_n{D}_n$ nonempty. Every choice from this intersection extends the prefix and preserves the invariant.

By [F6], make the recursively compatible selections from step 2.1. For each $n$, once the first $k_n$ active coordinates have been selected, they lie in the finite base of $C_n$ because its remaining section has positive mass. Fill any inactive coordinates with the product point supplied by [F4]. The resulting point lies in every $C_n$, contradicting ${\bigcap }_n{C}_n=\mathrm{\varnothing }$. Thus ${\mu }_0(C_n)\downarrow 0$.

Finite additivity plus continuity at the empty set gives countable additivity whenever a disjoint union remains a cylinder: apply it to the decreasing remainders. Hence ${\mu }_0$ is a premeasure.

Define ${\mu }_0({\pi }_F^{-1}(A))=({⨂}_{n\in F}{\mu }_n)(A)$. The finite product marginals are consistent, so [F1] applies.

By [F2], ${\mu }_0$ extends to a measure $\mu$ on the generated cylinder sigma-algebra. Since the empty cylinder is $E$ and has value $1$, $\mu$ is a probability measure.

Let $\nu$ be another probability measure with the stated marginals. The family $\mathcal{D}=\{A\in {\mathcal{C}}_{\mathbb{N}}:\mu (A)=\nu (A)\}$ is a lambda-system: it contains the whole space because both measures have mass one, is closed under relative complements of nested members, and is closed under increasing countable unions by continuity from below. The two measures agree on every cylinder, and cylinders are a pi-system by Finite-coordinate cylinders form a $\pi$-system, so [F3] gives ${\mathcal{C}}_{\mathbb{N}}\subseteq \mathcal{D}$. Hence $\mu =\nu$.

For $A_j\in {\mathcal{E}}_{n_j}$ and the distinct indices fixed above, [F1] gives $\mathbb{P}({\bigcap }_j\{X_{n_j}\in A_j\})={\prod }_j{\mu }_{n_j}(A_j)$; taking one coordinate gives the asserted law.

The right side is ${\prod }_j\mathbb{P}(X_{n_j}\in A_j)$, which is exactly [F2]. Since the finite family was arbitrary, all coordinates are independent.

Apply [F1] with $E_n=S$ and ${\mu }_n=\nu$ for every $n$, and write $X_n$ for its coordinate maps.

By [F2], every $X_n$ has law $\nu$ and every finite subfamily is independent; this is the required independent-copy sequence.

Choose Polish presentations $h_j:E_j\to P_j$ from [F1]. The product map $h={\prod }_{j<n}{h}_j$ is a bijection from $\prod E_j$ to $\prod P_j$.

A finite product of Polish spaces is Polish, and inverse images under $h$ of its Borel rectangles are exactly the finite product measurable rectangles. Therefore $h$ transports its Borel sigma-algebra to ${⨂}_{j<n}{\mathcal{E}}_j$, proving the claim.

The empty $P$ is immediate, so suppose $P\mathrm{\ne }\mathrm{\varnothing }$ and enumerate a countable dense set as $(x_m)$. For each $r\ge 1$, continuity from below chooses a finite union $U_r$ of $2^{-r-2}$-balls centered at the $x_m$ with $\mu (P\setminus U_r)<\epsilon 2^{-r-2}$. Countable choice makes these choices simultaneously.

The set $K_0={\bigcap }_r\overline{U_r}$ is closed. Countable subadditivity gives $\mu (P\setminus K_0)<\epsilon \mathrm{/}2$. For every scale, one of the finite closed-ball covers from step 1.1 covers $K_0$; choosing one point of $K_0$ from each nonempty member of that finite cover and doubling the radius gives a finite net with centres in $K_0$. Hence $K_0$ is totally bounded and [F3] makes it compact.

Every open $G$ contains a compact $K\subseteq G$ losing less than $\epsilon$: for $G=P$ use $K_0$; otherwise intersect $K_0$ with the increasing closed sets $\{x:d(x,P\setminus G)\ge 1\mathrm{/}n\}$. Their union is $K_0\cap G$, so [F2] gives one with the required loss.

Let $\mathcal{R}$ be the Borel sets which, for every $\delta >0$, have compact $K\subseteq A\subseteq G$ with $G$ open and $\mu (G\setminus K)<\delta$. Step 3.1 puts every open set in $\mathcal{R}$. The tight compact set from step 2.1 shows that complements remain in $\mathcal{R}$: from $K\subseteq A\subseteq G$, use $K_0\setminus G\subseteq P\setminus A\subseteq P\setminus K$ and bound the loss by $\mu (P\setminus K_0)+\mu (G\setminus K)$.

For pairwise disjoint $A_j\in \mathcal{R}$, continuity from below makes the measure of the tail of ${\bigcup }_j{A}_j$ arbitrarily small. A finite union of compact inner approximants handles that tail, while the union of the open outer approximants has loss bounded by the summable errors. Thus $\mathcal{R}$ is a lambda-system. Open sets are a pi-system generating the Borel sigma-algebra, so [F4] gives $\mathcal{B}(P)\subseteq \mathcal{R}$. The compact inner approximant for $A$ proves the statement.

By AC, choose once and for all, for every $i\in I$, a Polish presentation $h_i:E_i\to P_i$ witnessing that $(E_i,{\mathcal{E}}_i)$ is standard Borel. Pull the topology and a complete compatible metric of $P_i$ back to $E_i$. The finite product topologies now used below come from these same coordinate presentations, so every restriction map between them is continuous.

Let $C_n\downarrow \mathrm{\varnothing }$ be cylinders and suppose their cylinder masses stayed above $\eta >0$. Enumerate the countable union of their finite supports. For each $n$, choose a nested finite support $H_n$ that contains both the original support of $C_n$ and the first $n$ active coordinates, and view $C_n$ as a cylinder over $H_n$. In the product presentation fixed in step 1.1, inner-regularly replace its $H_n$-base by a compact base, losing less than $\eta 2^{-n-2}$. Finite intersections of the lifted compact cylinders then have positive cylinder mass.

For each $N$, choose a point in the intersection of the first $N$ lifted compact cylinders from step 2.1. For fixed $n$, all points with $N\ge n$ have their $H_n$-coordinates in the compact $n$th base. Sequential compactness successively supplies a subsequence converging on $H_1$, a further subsequence converging on $H_2$, and so on; take the diagonal subsequence. Because every restriction ${\prod }_{i\in H_{n+1}}{E}_i\to {\prod }_{i\in H_n}{E}_i$ is continuous by step 1.1, the successive limits restrict to the earlier limits. They therefore define one point on the union of the active coordinates. Each compact base is closed, so this point lies in every lifted compact cylinder.

Use AC to fill the inactive coordinates. The resulting point lies in every $C_n$, a contradiction. Hence the cylinder law is continuous at the empty set and is a premeasure.

By [F4] it extends to ${\mathcal{C}}_I$ and has total mass one. Two such extensions agree on all cylinders; these are a pi-system, so Dynkin's pi-lambda theorem gives uniqueness on ${\mathcal{C}}_I$, and nowhere larger.

For finite $F$, define ${\nu }_F={⨂}_{i\in F}{\mu }_i$. The rectangle formula shows that projecting ${\nu }_G$ to $F\subseteq G$ gives ${\nu }_F$, so the family is consistent.

Apply [F2] to $({\nu }_F)$. Its conclusion is exactly the stated measure and its cylinder-sigma uniqueness; it does not assert a measure on $\mathcal{P}({\prod }_i{E}_i)$.

Each coordinate map is measurable because inverse images of its measurable sets are one-coordinate cylinders. Thus $(X_i)$ is a process.

For finite $F$, $(X_i{)}_{i\in F}={\pi }_F$, so its pushforward law is ${\mu }_F$ by [F1]. This is precisely the finite-dimensional-law convention.

On $\mathcal{D}=\{C\in {\mathcal{C}}_I:\mu (C)=\nu (C)\}$, complements and disjoint countable unions preserve equality because both measures are probabilities. Thus $\mathcal{D}$ is a lambda-system.

Equal finite-dimensional marginals put every cylinder in $\mathcal{D}$. By [F1] and [F2], ${\mathcal{C}}_I\subseteq \mathcal{D}$, hence $\mu =\nu$.

Let $\mathcal{D}$ be the events determined by some countable coordinate set. Complements preserve the same support. For a countable family in $\mathcal{D}$, countable choice selects one countable support for each event, and [F2] makes their union countable; that union determines the union of the events. Thus $\mathcal{D}$ is a sigma-algebra.

Every finite-coordinate cylinder belongs to $\mathcal{D}$, using its finite support. By generated-sigma minimality and [F1], ${\mathcal{C}}_I\subseteq \mathcal{D}$.