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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

$$s_k(x):=\frac{x_1+\cdots+x_k}k.$$

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?

$X$ 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.

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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.

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    $\begingroup$ This is also the statement that almost every $x \in [0,1]$ is simply normal in base $2$. $\endgroup$
    – Robert Israel
    Commented Aug 16, 2019 at 15:27

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