Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated, in the sense of being badly mixed up with their complements; indeed, their existence is a non-trivial consequence of the axiom of choice.
Non-measurable sets in a Euclidean space are necessarily complicated. This was in the sense of being badly mixed up with their complements. The Lebesgue measure cannot be defined consistently on Non-measurable sets in a Euclidean space. Their existence is a non-trivial consequence of the axiom of choice. This is indeed.