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2011-10-23T17:21:11Z

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{{Unreferenced|date=December 2009}}
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 set|open]]" and "approximately [[Closed set|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 [[Wikt:finite|finite]] Lebesgue measure, then ''C'' can be chosen to be [[compact space|compact]] (i.e. – by the [[Heine–Borel theorem]] – closed and [[Bounded set|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'':

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

==See also==
* [[Radon measure]]

{{DEFAULTSORT:Regularity Theorem For Lebesgue Measure}}
[[Category:Theorems in measure theory]]

Integral representation theorem for classical Wiener space - Revision history

en>Geometry guy: subcat