Let $F = \{x\in[0,1]: x \text{ does not contain the digit 5}\}$

At first it's obvious that $1/2=0.5\in F$, but the problem arises when you write $1/2=0.499...\notin F$, which leads to a contradiction.

How do you measure such a set $F$ with this ambiguity? (This can be done as seen in [this](https://math.stackexchange.com/questions/2154298/measure-of-set-of-numbers-in-0-1-with-their-decimal-expansions-not-containin) question). Is it because the set of numbers that can be represented in such a non-unique way has measure $0$?