Regularity theorem for Lebesgue measure

In mathematics, the regularity theorem for Lebesgue measure is a result in measure theory that states that Lebesgue measure on the real line is a regular measure. Informally speaking, this means that every Lebesgue-measurable subset of the real line is "approximately open" and "approximately closed".

Statement of the theorem

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

• C is closed; and
• U is open; and
• C ⊆ A ⊆ U; and
• the Lebesgue measure of U \ C is strictly less than ε.

Moreover, if A has finite Lebesgue measure, then C can be chosen to be compact (i.e. – by the Heine–Borel theorem – closed and bounded).

Corollary: the structure of Lebesgue measurable sets

If A is a Lebesgue measurable subset of R, then there exists a Borel set B and a null set N such that A is the symmetric difference of B and N:

$A = B \triangle N = \left( B \setminus N \right) \cup \left( N \setminus B \right).$