Let $Y$ denote the set of points of $[0,1)$ that have a unique binary expansion. Then $Y$ has a countable complement so $m(Y)=1$, where $m$ is the Lebesgue measure.
I have to confess that I do not know much about binary expansion and so I do not understand this assumption.
1.) I thought that every point in $[0,1)$ has a unique binary expansion. Why not?
2.) Why is the complement of $Y$ countable andwhy is then $m(Y)=1$?
Maybe you can help me to understand this.
Thank you
Best Answer
(1) For instance: $0.01111\cdots=0.1$ in binary.
(2) All nonunique expansions are of the above form. In particular, the reals in $[0,1)$ with nonunique binary expansions are those that admit finite binary expansions. There are countably many of those.
Countable sets have Lebesgue measure zero.