For any positive integer n, let E(n) denote n-dimensional Euclidean Space and let L(n) denote n-dimensional Lebesgue Measure on E(n). Take n=2 for simplicity. There are uncountably many convex subsets of E(2) which are neither open nor closed in E(2). For example, if the subset D of E(2) is an open disk, then the union of D and any subset of its boundary (with respect to E(2)) is convex. My first question is "Are all convex subsets of the Euclidean plane L(2)-measurable, if we allow L(2) to be infinite?". My second question is "If S is any closed non-convex subset of E(2) and if C(S) is the smallest convex subset of E(2) that contains S as a subset, is C(S) necessarily closed in E(2)?". I suspect that both questions are well known to have "Yes" answers, although I have been unable to find any mention of this in the literature to which I have access.
The first question has an affirmative answer; see this question on math.se: https://math.stackexchange.com/questions/207609/the-measurability-of-convex-sets
(This was the first Google hit for "convex set Lebesgue measurable").