 Solution

The map swaps each adjacent pair of binary digits: $({\omega }_1,{\omega }_2,{\omega }_3,{\omega }_4,\dots )\mapsto ({\omega }_2,{\omega }_1,{\omega }_4,{\omega }_3,\dots )$. Canonical expansions with no eventual all-ones tail remain canonical under this swap, and ${\theta }^2=\mathrm{id}$.

Under Lebesgue measure, any specified values of $m$ distinct binary digits have probability $2^{-m}$. Indeed they describe a union of dyadic intervals at the largest of the specified positions, with that total length. The preimage of a cylinder prescribing its first $m$ output digits prescribes $m$ distinct input digits and therefore has the same measure. The half-open dyadic intervals, with the empty set, form a pi-system generating the Borel sigma-algebra. Apply the preceding uniqueness result to obtainThe choice of binary expansion matters at dyadic endpoints for the map's pointwise definition; the chosen convention removes that ambiguity and those endpoints are null in the measure calculation.