 Solution

Each function $(x,y)\mapsto |R_n(x)-R_n(y)|$ is Borel measurable on the product square, by composition with its coordinate projections and arithmetic of measurable real functions. Its value is one exactly when the $n$th digits disagree. Therefore the partial disagreement countsare nonnegative Borel functions and increase to the extended-real function $f$. For every finite $a$, $\{f>a\}={\bigcup }_N\{f_N>a\}$ is Borel. HenceInfinite disagreement is allowed, so restricting the target to finite real values would lose part of the stated function.