PMEA and PMEA-sigma

Work in $\mathrm{Z}\mathrm{F}\mathrm{C}$, and write $\mathfrak{c}=2^{{\mathrm{\aleph }}_0}$ (The Axiom of Choice, Cardinal (initial ordinal) and cardinality). For a cardinal $\lambda$, put $2^{\lambda }=\{x:x:\lambda \to \{0,1\}\}$.

For finite $F\subseteq \lambda$ and $s:F\to \{0,1\}$, let $C(F,s)=\{x\in 2^{\lambda }:x{\mathrm{\mid }}_F=s\}$. The finite-cylinder sigma-algebra ${\mathcal{C}}_{\lambda }$ is generated by these sets. It equals the cylinder sigma-algebra of Coordinate maps, finite-coordinate cylinders, and the cylinder $\sigma$-algebra, since any subset of a finite binary coordinate space is a finite union of singleton assignments. In particular $C(\mathrm{\varnothing },\mathrm{\varnothing })=2^{\lambda }$.

Give $\{0,1\}$ its full power set and the probability assigning each point mass $1\mathrm{/}2$. This is standard Borel: the discrete metric is complete and the finite space itself is a countable dense set, and its Borel sets are all its subsets (Standard Borel spaces). Thus Arbitrary products of standard Borel probability spaces, with index set $\lambda$ and these factors, supplies a unique probability ${\mu }_{\lambda }$ on ${\mathcal{C}}_{\lambda }$ with ${\mu }_{\lambda }(C(F,s))=2^{-\mathrm{\mid }F\mathrm{\mid }}$. Indeed its finite marginals are the finite coin products; conversely these cylinder masses determine every such marginal by finite disjoint unions. This is the fair-coin product measure. For finite $\lambda$ it is normalized counting measure on $2^{\lambda }$; for $\lambda =0$ it is the unit mass on the singleton empty function.

A full extension is a probability measure $\nu :\mathcal{P}(2^{\lambda })\to [0,1]$ restricting to ${\mu }_{\lambda }$ on ${\mathcal{C}}_{\lambda }$. Here a probability measure has total mass $1$ and is countably additive in the sense of Measures on sigma-algebras. Its full domain makes it complete (Complete measure spaces), but completeness alone does not mean that every subset of the underlying space is measurable.

A full extension is $\mathfrak{c}$-additive if for every pairwise disjoint family $(A_j{)}_{j\in J}$ with $\mathrm{\mid }J\mathrm{\mid }<\mathfrak{c}$, $\nu \left({\bigcup }_{j\in J}{A}_j\right)={\sum }_{j\in J}\nu (A_j),$ where the sum denotes the supremum of the finite subsums. Equivalently its null ideal is closed under unions of fewer than $\mathfrak{c}$ sets. To verify this equivalence, well-order a null family and disjointify it by subtracting earlier members; the resulting sets remain null, so the displayed identity makes their union null. Conversely, for a disjoint family in a probability space, at most $n$ members have mass at least $1\mathrm{/}n$ for each integer $n\ge 1$. Thus only countably many members have positive mass. Countable additivity sums those members, and the remaining fewer than $\mathfrak{c}$ null members have null union by the assumed null-ideal closure. This proves the identity. The cardinal bound is strictly less than $\mathfrak{c}$; no condition on increasing families of length $\mathfrak{c}$ is intended.

PMEA (the product measure extension axiom) asserts that for every cardinal $\lambda$, ${\mu }_{\lambda }$ has a $\mathfrak{c}$-additive full extension.

PMEA-$\sigma$ asserts that for every cardinal $\lambda$, ${\mu }_{\lambda }$ has a countably additive full extension.

PMEA implies PMEA-$\sigma$. The additional null-ideal closure in PMEA is the clause available to later consumers dealing with fewer than continuum many null sets. The product-measure corollary constructs ${\mu }_{\lambda }$ on ${\mathcal{C}}_{\lambda }$; it does not construct either full extension asserted by these axioms. Neither axiom asks for a two-valued measure on the index set $\lambda$.