 Lebesgue measure on the real line, R, is a regular measure.That is, for all Lebesgue-measurable subsets "A" of R, and "ε" > 0, there exist subsets "C" and "U" of R such that
Moreover, if "A" has Lebesgue measure, then "C" can be chosen to be compact (i.e.– by the Heine–Borel theorem – closed and bounded).