= 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 counts
$$
f_N(x,y)=\sum_{n=1}^N|R_n(x)-R_n(y)|
$$
are nonnegative Borel <functions> and increase to the extended-real <function> $f$. For every finite $a$, $\{f>a\}=\bigcup_N\{f_N>a\}$ is Borel. Hence
$$
\boxed{f:[0,1)^2\to[0,\infty]\text{ is nonnegative measurable}}.
$$
Infinite disagreement is allowed, so restricting the target to finite real values would lose part of the stated <function>.
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