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
Key properties include completeness, sigma-additivity, and translation invariance. Lebesgue measure is regular: for any measurable set
Lebesgue measurability underpins modern integration and analysis, providing a robust framework for measuring size and integrating