Lebesguemålebarhet

Lebesguemålebarhet refers to the concept of Lebesgue measurability in the context of the Lebesgue measure on Euclidean space. In measure theory, the Lebesgue measure m is a translation-invariant, complete measure that extends the intuitive notions of length, area, and volume. A set E ⊆ R^n is Lebesgue measurable if it satisfies Carathéodory’s criterion: for every A ⊆ R^n, m*(A) = m*(A ∩ E) + m*(A ∩ E^c), where m* denotes Lebesgue outer measure. This criterion identifies a sigma-algebra of measurable sets and yields a well-behaved notion of size.

Lebesgue outer measure is defined on all subsets of R^n by taking the infimum of the total lengths (or volumes) of countable coverings by open intervals (or boxes). The Lebesgue measurable sets form a sigma-algebra, and the Lebesgue measure m is defined on this sigma-algebra, extending the ordinary length of intervals and the measure of Borel sets. All Borel sets are Lebesgue measurable, and the measure is complete: every subset of a null set has Lebesgue measure zero and is measurable.

Key properties include completeness, sigma-additivity, and translation invariance. Lebesgue measure is regular: for any measurable set E, m(E) equals the supremum of the measures of compact subsets of E and also equals the infimum of the measures of open sets containing E. Examples illustrate the scope: the Cantor set has Lebesgue measure zero, while the existence of non-measurable sets—such as certain Vitali sets—shows that not every subset of R^n is Lebesgue measurable.

Lebesgue measurability underpins modern integration and analysis, providing a robust framework for measuring size and integrating a wide class of functions.