Effectively open sets in Cantor space

Cantor space $2^{\omega }$ is the set of infinite binary sequences. For $\sigma \in \{0,1{\}}^{\ast }$, let $[\sigma ]=\{X:\sigma \text{ is a prefix of }X\}$. Its choice-free fair-coin content is $2^{-\mathrm{\mid }\sigma \mathrm{\mid }}$, as in Binary-sequence cylinders and fair-coin content. The choice-free fair-coin open content ${\mu }_o$ of Choice-free fair-coin content on open subsets of binary sequence space assigns this same value to $[\sigma ]$ and is countably subadditive on open sets. Assuming countable choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), ${\mu }_o$ agrees on open sets with the unique Borel fair-coin probability of Fair-coin measure on binary sequences.

A set $W\subseteq \{0,1{\}}^{\ast }$ is computably enumerable when it is the range of a partial computable enumeration procedure (equivalently, some algorithm prints exactly its members, repetitions allowed), in the partial-computability convention of Computable and partial computable functions. An effectively open set is a union ${\bigcup }_{\sigma \in W}[\sigma ]$ for such a $W$. String conventions are those of Computation alphabets, words, the empty word, and ${\mathrm{\Sigma }}^{\ast }$. The definition of effective openness and the open-content value ${\mu }_o(U)$ require no choice principle.