= Solution
The <set> of finitely disagreeing pairs is $\bigcup_{N\ge0}F_N$, where $F_N$ consists of pairs whose digits agree at every position beyond $N$. For $M>N$, requiring agreement just at positions $N+1,\ldots,M$ 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 gives
$$
\boxed{\lambda^2\bigl(f^{-1}([0,\infty))\bigr)=0}.
$$
Thus almost every pair disagrees in infinitely many digits. Dyadic endpoints, treated with the stipulated expansion convention, do not change any rectangle measure.
Back to article page
