A consistent family of finite-dimensional distributions

Let $(E_i,{\mathcal{E}}_i{)}_{i\in I}$ be measurable spaces. A family $({\mu }_F{)}_{F⋐I}$, indexed by all finite subsets of $I$, is a consistent family of finite-dimensional distributions when each ${\mu }_F$ is a probability measure on ${⨂}_{i\in F}{\mathcal{E}}_i$, with the finite unordered product sigma-algebra of Coordinate maps, finite-coordinate cylinders, and the cylinder $\sigma$-algebra, and for every pair of finite sets $F\subseteq G\subseteq I$ the pushforward of ${\mu }_G$ under the restriction $p_{G,F}(x)=x{\mathrm{\mid }}_F$ equals ${\mu }_F$; equivalently, ${\mu }_G(p_{G,F}^{-1}(A))={\mu }_F(A)$ for every $A\in {⨂}_{i\in F}{\mathcal{E}}_i$. The restriction is measurable since inverse images of coordinate generators are coordinate generators. The empty support carries the unique probability measure on the singleton empty product.

Reordering means choosing a different enumeration of the same finite set $F$. If $e:[n]\to F$ is an enumeration, its displayed ordered law is the pushforward of ${\mu }_F$ under $r_e(x)=(x_{e(0)},\dots ,x_{e(n-1)})$. For another enumeration $e^{\mathrm{\prime }}$, the coordinate permutation $r_{e^{\mathrm{\prime }}}\circ r_e^{-1}$ pushes this ordered law to the one displayed using $e^{\mathrm{\prime }}$. This is automatic from these definitions and does not require equal laws on different subsets of $I$, or invariance under permutations of a fixed ordered product. In particular, consistency is data about every finite joint law, not merely the one-coordinate marginals.