NationMaster - Encyclopedia: Regularity theorem for Lebesgue measure

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". Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ... In mathematics, the real line is simply the set of real numbers. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related branches of mathematics, a closed set is a set whose complement is open. ...

Statement of the theorem

Lebesgue measure is a regular measure. That is, for all Lebesgue-measurable subsets $A$ of the real line, and , there exist subsets $C$ and $U$ of the real line such that

$C$ is closed;
$U$ is open;
; and
the Lebesgue measure of is strictly less than .

Moreover, if $A$ has finite Lebesgue measure, then $C$ can be chosen to be compact (i.e. closed and bounded). In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ...

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 $N$ : In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space is either of two σ-algebras on a topological space X: The minimal σ-algebra containing the open sets. ... In measure theory, a null set is a set that is negligible for the purposes of the measure in question. ... In mathematics, the symmetric difference of two sets is the set of elements which are in one of either set, but not in both. ...

Statement of the theorem GA_googleFillSlot("encyclopedia_square");

Corollary: the structure of Lebesgue measurable sets

See also

Statement of the theorem