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

The set of finitely disagreeing pairs is ${\bigcup }_{N\ge 0}{F}_N$, where $F_N$ consists of pairs whose digits agree at every position beyond $N$. For $M>N$, requiring agreement just at positions $N+1,\dots ,M$ defines a union of dyadic rectangles of total area $2^{-(M-N)}$: among the $2^{2M}$ pairs of length-$M$ prefixes, exactly $2^{M+N}$ meet these $M-N$ matching constraints, each rectangle having area $2^{-2M}$.

Since $F_N$ is contained in each of these finite-prefix agreement sets, its measure is at most $2^{-(M-N)}$ for every $M$, hence zero. Countable subadditivity now givesThus almost every pair disagrees in infinitely many digits. Dyadic endpoints, treated with the stipulated expansion convention, do not change any rectangle measure.