Consider the unit interval $[0,1]$, and by digits of $x\in[0,1]$ I mean its binary digits after the separator with no 1-period.
If $x\_1,x\_2,x\_3,...$ are the digits of $x$, then consider the *$k$-th average*


$$s\_k(x):=\frac{x\_1+\cdots+x\_k}k.$$



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> **Question:** What is the Lebesgue measure of the set $$X:=\{x\in[0,1]\mid \text{$s\_k(x)$ converges for $k\to\infty$}\}.$$ Is it maybe a null-set or the complement of a null-set?
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$X$ is uncountable, as any $y\in[0,1]$ can be obtained as limit of $s\_k(x)$ for infinitely many distinct $x\in[0,1]$.
We have that $X$ is the union of the following sets:


$$X(y):=\{x\in[0,1]\mid \lim\_{k\to\infty} s\_k(x) = y\},\quad \text{for some }y\in[0,1].$$


I believe that any of these is a null-set.



Measure of real numbers with converging average over binary digits
The set $\{x\in[0,1]:\lim\_k s\_k(x)=\frac12\}$ is the complement of a null set. This is an instance of the strong law of large numbers.


