In my self-study, I came across the following interesting little exercise:
Let $A \subseteq [0,1]$ be the set of numbers $x$ between zero and one such that the digit 3 appears before the digit 2 in the decimal expansion of $x$. Prove that $A$ is measurable and find its Lebesgue measure directly.
I am aware of a "sleight-of-hand" probability theory way for computing the Lebesgue measure of this set, and from there one might conclude that $A$ is measurable without being rigorous, but I am looking for a proof that proceeds by doing each of the following with sufficient rigor: verifying the measurable claim FIRST, and then evaluating the Lebesgue measure.