# Measure of Cesaro summable sequences

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.

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 $$1$$.