I have seen a proof showing that there are subsets of $\mathbb{R}$ which are not Lebesgue measurable. If I recall correctly it uses the axiom of choice. My first question is, are there sensible sets that are not measurable, i.e. something I can actually be given a description of and not just be shown to exist, or at least, has somebody found one? Do all such sets require the axiom of choice, and does the existence of such sets imply the axiom of choice?
Thanks for any ideas, or any good references (preferably online)!