The fair-coin one-sided shift preserves measure and is mixing

For a cylinder C prescribed on F, ${\sigma }^{-1}C$ imposes the same values on $F+1$. It is a cylinder with the same number of fixed coordinates and the same mass. Inverse images of prefix cylinders are open, so sigma is continuous and hence Borel measurable. Cylinders together with the empty set form a generating pi-system containing Omega. Since its mass is one, the preservation criterion applies; the completion clause then gives completed preservation.

If C and H prescribe finite coordinate sets F and G, then ${\sigma }^{-n}H$ prescribes G+n. For all sufficiently large n these are disjoint from F, so their intersection is a cylinder prescribing $\mathrm{\mid }F\mathrm{\mid }+\mathrm{\mid }G\mathrm{\mid }$ coordinates. Its mass is $2^{-\mathrm{\mid }F\mathrm{\mid }-\mathrm{\mid }G\mathrm{\mid }}=p(C)p(H)$. Empty sets give zero correlations. The mixing criterion now proves mixing for arbitrary Borel pairs.

For two completed sets replace each by a Borel core modulo a Borel null cover, as supplied by the completion construction. The symmetric difference of the intersection and its Borel version lies in the union of the first null cover and the nth pullback of the second. Preservation makes that union null. Correlations and marginal masses therefore agree exactly with their Borel versions for each n, proving completed mixing. Finally the mixing implication proves ergodicity for both probability spaces. Countable choice is inherited from the fair-coin extension and completion; coordinate calculations use only finite counting.