Binary-sequence cylinders and fair-coin content

Put $\mathrm{\Omega }=\{0,1{\}}^{{\mathbb{N}}_0}$, the set of functions from nonnegative integers to $\{0,1\}$. For a finite set $F\subseteq {\mathbb{N}}_0$ and a function $a:F\to \{0,1\}$ the cylinder $[a{]}_F$ consists of x with $x_j=a(j)$ for every j in F. The empty prescription gives all of Omega. Every cylinder is nonempty by filling unspecified coordinates with zero.

The cylinder algebra $\mathcal{C}$ is the set of finite unions of cylinders, including the empty union. It is an algebra in the sense of Algebras of subsets: any finite list of prescriptions can be refined to their finite coordinate union G; the $2^{\mathrm{\mid }G\mathrm{\mid }}$ complete prescriptions on G are disjoint nonempty atoms partitioning Omega, and union, intersection and complement of unions of these atoms again are such unions.

Give each G-atom mass $2^{-\mathrm{\mid }G\mathrm{\mid }}$. If A is a union of m distinct G-atoms, define its fair-coin content by $p_0(A)=m{2}^{-\mathrm{\mid }G\mathrm{\mid }}$. This is independent of G and its representation. Enlarging G to H splits each atom into exactly $2^{\mathrm{\mid }H\mathrm{\mid }-\mathrm{\mid }G\mathrm{\mid }}$ atoms, leaving its mass unchanged. Two representations agree after refining to their coordinate union; because all refined atoms are nonempty, the same subset A selects precisely the same atoms in both. Common refinement also proves finite additivity on disjoint sets. In particular $p_0(\mathrm{\varnothing })=0$, $p_0(\mathrm{\Omega })=1$, and $p_0([a{]}_F)=2^{-\mathrm{\mid }F\mathrm{\mid }}$. All constructions here involve finite coordinate sets and are choice-free.