Lebesguemessbare

Lebesgue-measurable (Lebesgue-messbare) is a central concept in measure theory referring to subsets of Euclidean space and to functions that are compatible with Lebesgue measure. It provides a framework for integration that extends beyond Riemann theory and handles a wider class of sets and functions.

For sets, a subset E of R^n is Lebesgue measurable if it satisfies Carathéodory’s criterion: for every A ⊆ R^n, the Lebesgue outer measure m*(A) equals m*(A ∩ E) plus m*(A ∩ E^c). Equivalently, Lebesgue measurable sets form the Lebesgue σ-algebra, generated by open sets, and include all Borel sets. The Lebesgue measure m defined on this σ-algebra is complete (every subset of a measure-zero set is measurable), translation-invariant, and σ-finite on R^n.

For functions, a function f: R^n → R (or extended real) is Lebesgue measurable if the preimage of every open (or equivalently every Borel) set is Lebesgue measurable. Simple functions, which take only finitely many values on measurable sets, can approximate measurable functions from below or above, establishing the foundation of Lebesgue integration. Continuous functions are Lebesgue measurable, as are many naturally arising functions in analysis.

Lebesgue measure and measurability enable powerful convergence and integration theorems, such as the dominated convergence theorem, monotone convergence, and Fubini’s theorem. They also provide a robust setting where pathological issues present in Riemann integration can be managed, and where notions of almost everywhere equality and null sets play a fundamental role in analysis.