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# 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".

tatement 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 "&epsilon;" &gt; 0, there exist subsets "C" and "U" of R such that
* "C" is closed; and
* "U" is open; and
* "C" &sube; "A" &sube; "U"; and
* the Lebesgue measure of "U" "C" is strictly less than "&epsilon;".Moreover, if "A" has finite Lebesgue measure, then "C" can be chosen to be compact (i.e. &mdash; by the Heine-Borel theorem &mdash; 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":

:$A = B riangle N = left\left( B setminus N ight\right) cup left\left( N setminus B ight\right).$

ee also

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