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In mathematics, the regularity theorem for Lebesgue measure is a result that, informally speaking, shows that every Lebesgue-measurable subset of the real line is "approximately open" and "approximately closed".

[edit] Statement of the theorem

Lebesgue measure is a regular measure. That is, for all Lebesgue-measurable subsets <math>A</math> of the real line, and <math>\varepsilon > 0</math>, there exist subsets <math>C</math> and <math>U</math> of the real line such that

• <math>C</math> is closed;
• <math>U</math> is open;
• <math>C \subseteq A \subseteq U</math>; and
• the Lebesgue measure of <math>U \setminus C</math> is strictly less than <math>\varepsilon</math>.

Moreover, if <math>A</math> has finite Lebesgue measure, then <math>C</math> can be chosen to be compact (i.e. closed and bounded).

[edit] Corollary: the structure of Lebesgue measurable sets

If <math>A</math> is a Lebesgue measurable subset of the real line, then there exists a Borel set <math>B</math> and a null set <math>N</math> such that <math>A</math> is the symmetric difference of <math>B</math> and <math>N</math>:

<math>A = B \triangle N = \left( B \setminus N \right) \cup \left( N \setminus B \right).</math>

[edit] See also

• Radon measure