# Regularity theorem for Lebesgue measure

In

mathematics , the**regularity theorem for Lebesgue measure**is a result inmeasure theory that states thatLebesgue measure on thereal line is aregular 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 "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" hasfinite Lebesgue measure, then "C" can be chosen to be compact (i.e. — by theHeine-Borel theorem — closed and bounded).**Corollary: the structure of Lebesgue measurable sets**If "A" is a Lebesgue measurable subset of

**R**, then there exists aBorel set "B" and anull set "N" such that "A" is thesymmetric difference of "B" and "N"::$A\; =\; B\; riangle\; N\; =\; left(\; B\; setminus\; N\; ight)\; cup\; left(\; N\; setminus\; B\; ight).$

**ee also***

Radon measure

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