Let $S= \{x \in [0,1 ]: \lim_{n \to \infty } \frac {1 } {n } \sum _{i=1 } ^n d _i (x) = 2/3 \} $, where $d _i(x) $ gives the ith digit of the infinite base-2 expansion of $x $. What is the Lebesgue measure of $S $?
First I want to prove that $S $ is a measurable set, and secondly determine its measure.
$d _i(X) : [0,1 ]\mapsto \{0,1 \} $,
$d _1^{-1 } (0)= [0,1)$
$d _1^{-1 } (1)=\{1\} $
$d _2^{-1 } (0) = [0,\frac {1 } {2 } )$...
So that $d _i $ is a measurable map? Now the sum is nondecreasing and bounded, hence converges pointwise. Thus the sum is a measurable map? Hence $S $ is measurable?
If my reasoning is correct, then what is the measure?
Thanks in advance!