Decimal representation of the set [0,1)

Question

I have encountered the next statement in statistics lecture (translated from german): "From the analysis you know that all but a countable number of $$w ∈ [0, 1)$$ represent a unique decimal representation $$ω = 0.x_1x_2x_3...$$ for the countably many exceptions we choose that representation without period 9". That sounds very counterintuitive and unprovable to me, is this true?

Answer 1

This is true if “represent a” becomes “is represented by”. The exceptions are the numbers of $(0,1)$ which can be written with finitely many digits: $x=0.d_1d_2\ldots d_{n-1}d_n$ (with $d_n\in\{1,2,3,4,5,6,7,8,9\}$), because each such number can be also written as$$0.d_1d_2\ldots d_{n-1}(d_n-1)999\ldots$$For instance, $0.23=0.22999\ldots$ All other numbers have a single decimal expansion.