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.

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. 


I would say: The interior is an open set, so is measurable. The boundary (a smooth Jordan curve) has measure zero, so every subset of it is measurable. Our set is the union of the interior and a subset of the boundary, so it is also measurable. 


Let $C$ be a Jordan curve, and suppose that (the range of) $C$ has measure zero. The interior of $C$ is an open set and the given set (whose boundary is $C$) then differs from an open set by a set of measure zero (a subset of the range of $C$), so it is measurable. The question then boils down to, what conditions on $C$ suffice for its range to be of measure zero. It is not hard to see that a bound $M$ on the derivative of $C$ suffices, for then we can cover the range of $C$ with $2\pi/\epsilon$ disks of radius $\epsilon/(2\pi M)$, whose total area is less than $\epsilon$. If the Jordan curve is $C^1$ we have such a bound. However, this approach gets harder if we don't assume $C$ is $C^1$. So here's another attack: let $C$ be rectifiable, i.e. the limit of polygons of bounded length $L$. Then for each $\epsilon > 0$, the range of $C$ is contained in a union of rectangles, one about each side of an approximating polygon, where each rectangle has height $\epsilon/L$ and the total width of the rectangles is at most $L$. Since $\epsilon$ is arbitrary, the measure of a rectifiable Jordan curve is zero. Rectifiable is more general than $C^1$ so this improves the previous paragraph. I don't know whether there exists a differentiable but not rectifiable Jordan curve. 

