[proofplan] We first define a probability content on finite-coordinate cylinder sets by measuring the base of the cylinder using the prescribed finite-dimensional law. The consistency hypothesis makes this assignment independent of the finite coordinate representation of the same cylinder. The only substantial point is countable additivity on the cylinder algebra; for standard Borel state spaces this is the standard-Borel cylinder premeasure lemma, proved by compact approximation after representing the standard Borel space inside a Polish space. Caratheodory's [extension theorem](/theorems/59) then gives a [probability measure](/page/Probability%20Measure) on the product $\sigma$-algebra, and the defining values on cylinders identify all finite-dimensional pushforward laws. [/proofplan] [step:Define the cylinder algebra and its set function] For each finite subset $F\subset T$, define the coordinate projection \begin{align*} \pi_F:E_T\to E^F \end{align*} by $\pi_F(x)=(x(t))_{t\in F}$ for every $x\in E_T$. Let $\mathcal C$ denote the collection of finite-coordinate cylinder sets in $E_T$: \begin{align*} \mathcal C:=\{\pi_F^{-1}(A): F\subset T \text{ finite and } A\in \mathcal E^{\otimes F}\}. \end{align*} The class $\mathcal C$ is an algebra of subsets of $E_T$. Indeed, complements are preserved because \begin{align*} E_T\setminus \pi_F^{-1}(A)=\pi_F^{-1}(E^F\setminus A), \end{align*} and finite intersections are preserved because, for finite $F,G\subset T$ and $A\in\mathcal E^{\otimes F}$, $B\in\mathcal E^{\otimes G}$, \begin{align*} \pi_F^{-1}(A)\cap \pi_G^{-1}(B)=\pi_{F\cup G}^{-1}\left(\pi_{F\cup G,F}^{-1}(A)\cap \pi_{F\cup G,G}^{-1}(B)\right). \end{align*} The set on the right is measurable in $\mathcal E^{\otimes (F\cup G)}$, since both finite-coordinate projections are measurable. Define a set function \begin{align*} \nu_0:\mathcal C\to [0,1] \end{align*} as follows. If $C\in\mathcal C$ is represented as $C=\pi_F^{-1}(A)$ with $F\subset T$ finite and $A\in\mathcal E^{\otimes F}$, set \begin{align*} \nu_0(C):=\mu_F(A). \end{align*} [/step] [step:Show that the cylinder value is independent of the representation] We verify that $\nu_0$ is well-defined. Suppose \begin{align*} \pi_F^{-1}(A)=\pi_G^{-1}(B) \end{align*} for finite subsets $F,G\subset T$, with $A\in\mathcal E^{\otimes F}$ and $B\in\mathcal E^{\otimes G}$. Put $H:=F\cup G$. Then \begin{align*} \pi_H^{-1}\left(\pi_{H,F}^{-1}(A)\right)=\pi_F^{-1}(A)=\pi_G^{-1}(B)=\pi_H^{-1}\left(\pi_{H,G}^{-1}(B)\right). \end{align*} The projection $\pi_H:E_T\to E^H$ is surjective: if $T=\varnothing$ then $H=\varnothing$ and $\pi_H$ is the identity map on the one-point product, while if $T\neq\varnothing$ the statement assumes $E\neq\varnothing$, so any element of $E^H$ extends to an element of $E_T$ by choosing values in $E$ on $T\setminus H$. Therefore \begin{align*} \pi_{H,F}^{-1}(A)=\pi_{H,G}^{-1}(B). \end{align*} Using the consistency hypothesis for $F\subset H$ and $G\subset H$, we obtain \begin{align*} \mu_F(A)=\mu_H(\pi_{H,F}^{-1}(A))=\mu_H(\pi_{H,G}^{-1}(B))=\mu_G(B). \end{align*} Thus $\nu_0(C)$ does not depend on the chosen finite-coordinate representation. [guided] The possible ambiguity is that a cylinder may be described using more coordinates than are actually needed. For example, the same subset of $E_T$ might be written as a condition on coordinates in $F$ or as a condition on coordinates in a larger finite set. The consistency assumption is exactly what makes the assigned measure invariant under this refinement. Assume that \begin{align*} \pi_F^{-1}(A)=\pi_G^{-1}(B) \end{align*} with $F,G\subset T$ finite, $A\in\mathcal E^{\otimes F}$, and $B\in\mathcal E^{\otimes G}$. We compare both descriptions on the common finite coordinate set $H:=F\cup G$. The projection maps \begin{align*} \pi_{H,F}:E^H\to E^F,\qquad \pi_{H,G}:E^H\to E^G \end{align*} allow us to rewrite the two bases as subsets of $E^H$. Pulling them back to $E_T$ gives \begin{align*} \pi_H^{-1}\left(\pi_{H,F}^{-1}(A)\right)=\pi_F^{-1}(A) \end{align*} and \begin{align*} \pi_H^{-1}\left(\pi_{H,G}^{-1}(B)\right)=\pi_G^{-1}(B). \end{align*} By the assumed equality of cylinders, \begin{align*} \pi_H^{-1}\left(\pi_{H,F}^{-1}(A)\right)=\pi_H^{-1}\left(\pi_{H,G}^{-1}(B)\right). \end{align*} The map $\pi_H:E_T\to E^H$ is surjective. If $T=\varnothing$, then $H=\varnothing$ and $\pi_H$ is the identity map on the one-point product. If $T\neq\varnothing$, then $E\neq\varnothing$ by the theorem statement, so every finite coordinate assignment in $E^H$ can be extended to an element of $E_T$ by choosing values in $E$ on $T\setminus H$. Hence equality of the inverse images under $\pi_H$ forces equality of the sets in $E^H$: \begin{align*} \pi_{H,F}^{-1}(A)=\pi_{H,G}^{-1}(B). \end{align*} Now we use consistency. Since $F\subset H$, the hypothesis gives \begin{align*} (\pi_{H,F})_\#\mu_H=\mu_F, \end{align*} so \begin{align*} \mu_F(A)=\mu_H(\pi_{H,F}^{-1}(A)). \end{align*} Similarly, \begin{align*} \mu_G(B)=\mu_H(\pi_{H,G}^{-1}(B)). \end{align*} The two preimage sets inside $E^H$ are equal, so the two numbers are equal: \begin{align*} \mu_F(A)=\mu_G(B). \end{align*} Therefore the value assigned to a cylinder is independent of its representation. [/guided] [/step] [step:Verify finite additivity on disjoint cylinder decompositions] Let $C_1,\dots,C_m\in\mathcal C$ be pairwise disjoint cylinder sets, and suppose \begin{align*} C=\bigcup_{j=1}^m C_j\in\mathcal C. \end{align*} Choose finite subsets $F_1,\dots,F_m,F\subset T$ and measurable sets $A_j\in\mathcal E^{\otimes F_j}$, $A\in\mathcal E^{\otimes F}$ such that \begin{align*} C_j=\pi_{F_j}^{-1}(A_j),\qquad C=\pi_F^{-1}(A). \end{align*} Set \begin{align*} H:=F\cup F_1\cup \cdots \cup F_m. \end{align*} Define measurable subsets $D,D_1,\dots,D_m$ of $E^H$ by \begin{align*} D:=\pi_{H,F}^{-1}(A),\qquad D_j:=\pi_{H,F_j}^{-1}(A_j). \end{align*} The disjoint union identity in $E_T$ and the surjectivity of $\pi_H$ imply \begin{align*} D=\bigcup_{j=1}^m D_j, \end{align*} with the $D_j$ pairwise disjoint. Therefore finite additivity of the probability measure $\mu_H$ gives \begin{align*} \nu_0(C)=\mu_H(D)=\sum_{j=1}^m \mu_H(D_j)=\sum_{j=1}^m \nu_0(C_j). \end{align*} Thus $\nu_0$ is finitely additive on $\mathcal C$, and $\nu_0(E_T)=1$ because $E^\varnothing$ is the one-point product, $E_T=\pi_{\varnothing}^{-1}(E^\varnothing)$, and $\mu_{\varnothing}$ is the corresponding one-point probability measure. [/step] [step:Use the standard-Borel hypothesis to obtain a cylinder premeasure] We now use the standard-Borel cylinder premeasure lemma for projective families over standard Borel spaces: if $(E,\mathcal E)$ is standard Borel and a finitely additive probability content on the finite-coordinate cylinder algebra of $E_T$ has consistent finite-dimensional marginals, then it is countably additive on that cylinder algebra. The hypotheses of this lemma are satisfied here: $(E,\mathcal E)$ is standard Borel by assumption, the preceding steps constructed a finitely additive probability content $\nu_0:\mathcal C\to[0,1]$, and the finite-dimensional marginals are consistent by the stated projection hypothesis. Hence $\nu_0$ is a premeasure. The lemma is usually proved by representing $(E,\mathcal E)$ as a Borel subset of a Polish space and using compact approximation of finite-dimensional probability measures to rule out decreasing sequences of cylinders with empty intersection and positive limiting mass. For completeness, we spell out the precise consequence needed here. If $(C_n)_{n\ge1}$ is a decreasing sequence in $\mathcal C$ with \begin{align*} \bigcap_{n=1}^{\infty}C_n=\varnothing, \end{align*} then \begin{align*} \lim_{n\to\infty}\nu_0(C_n)=0. \end{align*} Since $\nu_0$ is finitely additive and $\nu_0(E_T)=1$, this continuity from above at the empty set is equivalent to countable additivity on the algebra $\mathcal C$. Hence $\nu_0$ is a premeasure on $\mathcal C$. [guided] Finite additivity alone is not enough for extension to a measure. The missing property is countable additivity on the cylinder algebra. A convenient equivalent form, for a finitely additive probability content, is continuity from above at the empty set: whenever $(C_n)_{n\ge1}$ is a decreasing sequence of sets in the algebra and the intersection is empty, the assigned masses must decrease to $0$. Here this is the deep point where the standard Borel assumption is used. The relevant standard result is the standard-Borel cylinder premeasure lemma: If $(E,\mathcal E)$ is standard Borel and a family of finite-dimensional probability measures $(\mu_F)_{F\subset T,\ F\text{ finite}}$ is projectively consistent, then the induced cylinder content on $E_T$ is countably additive on the cylinder algebra. We verify its hypotheses in this proof. The space $(E,\mathcal E)$ is standard Borel by the theorem statement. The family $(\mu_F)$ is projectively consistent because the theorem assumes $(\pi_{G,F})_\#\mu_G=\mu_F$ for all finite $F\subset G\subset T$. The preceding steps showed that this family induces a well-defined finitely additive probability content $\nu_0$ on the cylinder algebra $\mathcal C$. This result is not a formal consequence of finite additivity. Its proof uses the fact that standard Borel spaces can be realized as Borel subsets of Polish spaces. In finite-dimensional products, probability measures on standard Borel spaces admit enough regular approximation by compact sets inside a Polish realization. That compact approximation prevents the following obstruction: a decreasing sequence of cylinder conditions could have positive limiting finite-dimensional mass while having no point satisfying all conditions simultaneously. The compactness argument extracts compatible finite-coordinate points and then assembles them into an element of $E_T$, contradicting emptiness of the intersection. Applying this lemma to the finitely additive content $\nu_0$ constructed above gives the exact property needed: for every decreasing sequence $(C_n)_{n\ge1}$ in $\mathcal C$ satisfying \begin{align*} \bigcap_{n=1}^{\infty}C_n=\varnothing, \end{align*} we have \begin{align*} \lim_{n\to\infty}\nu_0(C_n)=0. \end{align*} Because $\nu_0$ is finitely additive and has total mass $1$, this continuity from above is equivalent to countable additivity on $\mathcal C$. Therefore $\nu_0$ is a premeasure. [/guided] [/step] [step:Extend the premeasure to the product sigma-algebra] By the [Caratheodory extension theorem](/theorems/9707), applied to the premeasure $\nu_0$ on the algebra $\mathcal C$ of subsets of $E_T$, there exists a measure \begin{align*} \mathbb P:\sigma(\mathcal C)\to[0,1] \end{align*} such that \begin{align*} \mathbb P(C)=\nu_0(C) \end{align*} for every $C\in\mathcal C$. Since $\nu_0(E_T)=1$, the extension satisfies \begin{align*} \mathbb P(E_T)=1, \end{align*} so $\mathbb P$ is a probability measure. It remains to identify $\sigma(\mathcal C)$. By definition, $\mathcal E_T$ is the product $\sigma$-algebra generated by the coordinate maps $X_t:E_T\to E$, $X_t(x)=x(t)$. Equivalently, $\mathcal E_T$ is generated by all finite-coordinate cylinders $\pi_F^{-1}(A)$ with $F\subset T$ finite and $A\in\mathcal E^{\otimes F}$. Therefore \begin{align*} \sigma(\mathcal C)=\mathcal E_T. \end{align*} Thus $\mathbb P$ is a probability measure on $(E_T,\mathcal E_T)$. [/step] [step:Identify the finite-dimensional laws of the coordinate process] Let $F\subset T$ be finite. We prove that \begin{align*} (\pi_F)_\#\mathbb P=\mu_F. \end{align*} For every $A\in\mathcal E^{\otimes F}$, the set $\pi_F^{-1}(A)$ belongs to $\mathcal C$, so the definition of the extension gives \begin{align*} (\pi_F)_\#\mathbb P(A)=\mathbb P(\pi_F^{-1}(A))=\nu_0(\pi_F^{-1}(A))=\mu_F(A). \end{align*} Thus the two probability measures agree on every measurable subset of $(E^F,\mathcal E^{\otimes F})$, and hence \begin{align*} (\pi_F)_\#\mathbb P=\mu_F. \end{align*} Since $\pi_F(x)=(X_t(x))_{t\in F}$ for every $x\in E_T$, this says exactly that the coordinate process $(X_t)_{t\in T}$ has finite-dimensional distribution $\mu_F$ on the coordinate set $F$. As $F\subset T$ was arbitrary, the theorem follows. [/step]