Fair-coin measure on binary sequences

The prefix cylinders form a countable base: length m has exactly $2^m$ possible prefixes, enumerated by the binary integers from 0 to $2^m-1$, and the pairs of length and index admit a diagonal enumeration. Every cylinder is open, and every open set is the union of the subfamily of prefix cylinders contained in it. It follows that the sigma-algebra generated by the cylinder algebra is exactly the metric Borel sigma-algebra.

Apply the extension theorem to the premeasure p_0. It gives a Borel measure p agreeing with every cylinder mass and with $p(\mathrm{\Omega })=p_0(\mathrm{\Omega })=1$. Any other measure with these cylinder masses agrees on finite disjoint unions by finite additivity, hence on the entire algebra; finite-premeasure uniqueness makes it equal to p.

The completion theorem gives a complete probability extending p, since the whole-space mass remains one. Countable choice is used by the cited extension and completion constructions; the preceding finite-algebra and compactness arguments themselves were choice-free. This constructs this particular binary measure, not a general infinite product measure.