Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 1 / 26J / c

The chosen non-eventually-one binary expansion fixes the ambiguity at dyadic rationals. Its $n$th digit isIn particularThis is a finite union of Borel sets; the zero-digit preimage is its complement in $[0,1)$. Since $R_n$ only takes the values zero and one, every Borel preimage is a union of these two sets. Thus it is Borel measurable. The half-open intervals implement the terminating-zero convention at dyadic endpoints.