Measure of Cesaro summable sequences

Question

Loosely, I would like to know "how many" binary sequences are non-convergent in the sense of their average.

Let $\{0,1\}^{\mathbb N}$ denote the set of all binary sequences. This set is homeomorphic to the unit interval and has therefore Lebesgue measure 1.

Let $C :=\{a \in \{0,1\}^{\mathbb N}: \lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^N a_n\text{ exists}\}$, i.e. the set of sequences in $\{0,1\}^{\mathbb N}$ that are 1-Cesaro summable.

(1) Does $C$ have positive Lebesgue measure?

(2) Does $C$ have Lebesgue measure 1?

Any reference relating to this or related problems is welcome.

Answer 1

Almost every $x \in [0,1]$ has binary digits with Cesaro limit $1/2$.

This is due to Borel, around 1910 or something. He was the first to write that when the unit interval considered as a probability space, the binary digits are i.i.d random variables with Bernoulli $(1/2,1/2)$ distribution. So the statement above is a simple instance of the strong law of large numbers.

So not only does $C$ have measure $1$, the subset $$ C_{1/2} :=\left\{a \in \{0,1\}^{\mathbb N}: \lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^N a_n = \frac{1}{2}\right\} $$ has measure $1$.