What is the cardinality of a non-measurable set?

Question

We know that $|P( \Bbb{R} )|=|L (\Bbb{R} )|$ ( $L (\Bbb{R} )$ is the set of all Lebesgue-measurable sets).

Note that $L (\Bbb{R} ) \subsetneq P( \Bbb{R} )$.

What is the cardinality of non-measurable set? Is this set countable?

Answer 1

A standard argument to show that $\mathcal{L}(\mathbb R)$ has the same size as $\mathcal{P}(\mathbb R)$ is to note that the Cantor subset of $[0,1]$ has the same size as $\mathbb R$ and measure 0, so any of its subsets also has measure 0. You can use the same idea to find the size of $\mathcal P(\mathbb R)\setminus\mathcal L(\mathbb R)$: Fix a nonmeasurable subset $N$ of $[2,3]$, and note that $N\cup A$ is nonmeasurable for any $A$ subset of the Cantor set. (This was asked before, by the way, see here.)

The question of what sizes can nonmeasurable sets have is harder. Of course, any such set is uncountable. If $\kappa$ is the least possible size of a nonmeasurable set, then there are nonmeasurable sets of size $\tau$ for any $\tau$ with $\kappa\le\tau\le|\mathbb R|$, by the same argument as in the previous paragraph, so the problem is to see what one can say about $\kappa$ itself. It turns out that $\mathsf{ZFC}$ is not strong enough to give us much information: it is consistent that $\kappa=\aleph_1$ while $|\mathbb R|$ itself can be arbitrarily large. It is also consistent that $\kappa=|\mathbb R|$ and $\mathbb R$ can be as large as wanted. Other behaviors are also consistent.

This number $\kappa$ has been studied in the context of cardinal characteristics (or "cardinal invariants") of the continuum, where it is denoted $\mathrm{non}(\mathcal L)$. There are several survey articles containing more information. See for instance the chapters by Andreas Blass and by Tomek Bartoszynski in the Handbook of set theory.