Regularity theorem for Lebesgue measure

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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 $A$ of the real line, and $\varepsilon > 0$ , there exist subsets $C$ and $U$ of the real line such that

$C$ is closed;
$U$ is open;
$C \subseteq A \subseteq U$ ; and
the Lebesgue measure of $U \setminus C$ is strictly less than $\varepsilon$ .

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

[edit] Corollary: the structure of Lebesgue measurable sets

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