Given some set of natural numbers, we can sometimes define that set's asymptotic density. The asymptotic density of a set will always lie in the interval $[0,1]$.
Since we can map sets of natural numbers onto real numbers in $[0,1)$ via the binary expansion, and vice versa (always using the terminating binary expansion when possible), we can thus speak of the asymptotic density of a real number.
Then, for some density $0 \leq d \leq 1$, we can construct the set of all real numbers $\Bbb R_d$ with asymptotic density = $d$. We can also construct the set of real numbers where the asymptotic density is undefined, denoted here by $\Bbb R_\times$.
Here are my questions:
- What are the cardinalities of the individual $\Bbb R_d$?
- Are all of the $\Bbb R_d$ measurable? (And if so, presumably, all of measure zero?)
- What is the cardinality of the set $\Bbb R_\times$?
- Is the set $\Bbb R_\times$ measurable, and if so, what is its measure?
Partly I am curious which of these things are provable in ZFC, or which might be related to things like CH. I am also very interested in seeing how the picture might change if the axiom of choice is dropped.
By the normal number theorem, we have that the outer measure of $\mathbb{R}_{1\over 2}$ is $1$. Of course, this leaves open the question of whether $\mathbb{R}_{1\over 2}$ is actually measurable.
The measurability of $\mathbb{R}_{1\over 2}$ is a consequence of a much stronger fact: that $\mathbb{R}_{1\over 2}$ is Borel (in fact $\Pi^0_3$ - the same complexity as the set of normal numbers). This finishes the job since every Borel set is of course measurable.
In fact, every one of the sets you mention is Borel.