Lebesguemessbaren

Lebesguemessbaren is a German term used to describe objects in measure theory that are measurable with respect to the Lebesgue measure. In practice it refers to sets or functions that are compatible with the Lebesgue notion of size and integration.

The Lebesgue measure is a translation-invariant measure defined on subsets of Euclidean space R^n, extending ordinary concepts of length, area, and volume. It forms the basis of the Lebesgue integration theory and leads to a complete sigma-algebra of measurable sets, denoted typically as the Lebesgue sigma-algebra, which contains all Borel sets and more.

Lebesgue measurable sets. A subset A of R^n is Lebesgue measurable if it can be approximated, up to a set of Lebesgue measure zero, by a Borel set. Equivalently, A satisfies Carathéodory’s criterion: for every E ⊆ R^n, the outer measure m*(E) equals m*(E ∩ A) + m*(E \ A). From this, the Lebesgue measure m is defined on A and extended to all Lebesgue measurable sets. Open, closed, and countable unions of such sets are Lebesgue measurable; there exist non-measurable sets as well (e.g., a Vitali set) under choice axioms.

Lebesgue measurable functions. A function f: R^n → R is Lebesgue measurable if the preimage of every open set is Lebesgue measurable. Equivalently, for every α ∈ R, the set {x : f(x) > α} is Lebesgue measurable. Simple functions, indicator functions of Lebesgue measurable sets, and many common functions are Lebesgue measurable. Lebesgue measurability is central to integration and theorems such as the dominated convergence and Fubini’s theorem.

In summary, Lebesguemessbaren denotes objects compatible with Lebesgue measure, encompassing measurable sets and measurable functions essential to modern analysis.