I learned analysis a while ago, so let me define what I want. Suppose we have a set whose boundary is a closed Jordan curve that has a unique tangent at each point. Is it true that this set is (Lebesgue) measurable? What if we also suppose that the tangents change continuously? I am interested in any modification of the question which is known.



Are there non-measurable sets with smooth boundary?
As was pointed out, the question did not intend the set to be the interior of the curve. The question now boils down to whether a differentiable Jordan curve has measure zero. This follows from the Lebesgue density theorem, see e.g. <http://en.wikipedia.org/wiki/Lebesgue>'s\_density\_theorem.


If the curve is differentiable, then at each point of the curve the Lebesgue density is zero. By the Lebesgue density theorem, the measure of the curve must be zero.


