We know that the cardinality of a subset of $\mathbb{R}$ of positive Lebesgue measure is the continuum. What can we say about the cardinality of a subset of $\mathbb{R}$ of positive outer measure?
Cardinality of a subset of $\mathbb{R}$ of positive outer measure
measure-theoryset-theory
Best Answer
The cardinality is independent of the axioms of $\mathsf{ZFC}$.
Given the ideal $\mathcal N$ of Lebesgue null sets, we call the least cardinality of a subset of $\Bbb R$ that is not Lebesgue null (i.e. has positive outer measure) the uniformity number of $\mathcal N$, or $\mathrm{non}(\mathcal N)$.
This is one of the cardinal characteristics featured in CichoĊ's diagram, and as such its cardinality has been extensively studied. Assuming that we have a model of $\mathsf{ZFC}$, then the method of forcing can be used to prove the existence of models where $\mathrm{non}(\mathcal N)<2^{\aleph_0}$. Conversely, it can also be shown to be consistent that $\aleph_1<\mathrm{non}(\mathcal N)=2^{\aleph_0}$.
In other words, it is relatively consistent to $\mathsf{ZFC}$ that there are sets of positive outer measure of cardinality less than the continuum, but it is also relatively consistent that no such sets exist, even when the continuum hypothesis does not hold.