 Solution

Outside the countable set of dyadic rational numbers, the binary expansion $x={\sum }_{j\ge 1}{ϵ}_j(x)2^{-j}$ is unique. Multiplication by two modulo one deletes the first binary digit, soLet $g=1_{[1/2,1)}$. By part (b), the doubling map is a measure-preserving transformation and is ergodic for Lebesgue measure. Applying the Birkhoff ergodic theorem to this indicator function givesfor almost every $x$. The dyadic rationals have Lebesgue measure zero, so the choice of their two possible binary expansions does not affect the conclusion. Thus the limiting frequency of the digit one is $1/2$ almost everywhere.