Non-measurable set

Summary

A non-measurable set is a set to which a given measure cannot assign a size, because it lies outside the $\sigma$-algebra of measurable sets. The term most often refers to subsets of $\R^n$ that are not measurable for Lebesgue measure. Their existence shows that length, area and volume cannot be defined for all sets in a way that is countably additive and invariant under translations. Their construction needs some form of the axiom of choice. Solovay showed that ZF with only dependent choice cannot prove that they exist, if an inaccessible cardinal is consistent with ZFC.