= Solution
The map swaps each adjacent pair of binary digits: $(\omega_1,\omega_2,\omega_3,\omega_4,\ldots)\mapsto(\omega_2,\omega_1,\omega_4,\omega_3,\ldots)$. Canonical expansions with no eventual all-ones tail remain canonical under this swap, and $\theta^2=\operatorname{id}$.
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 obtain
$$
\boxed{\theta\text{ preserves Lebesgue measure on }[0,1).}
$$
The 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.
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