Borelmessbare

Borelmessbare is the German term for Borel-measurable, used to describe functions or sets that are measurable with respect to the Borel sigma-algebra on a topological space. In mathematics, this concept plays a central role in measure theory and probability theory.

Definition: Let X be a topological space with its Borel sigma-algebra B(X), generated by the open sets, and let Y be a measurable space with sigma-algebra B(Y). A function f: X → Y is Borelmessbare if for every Borel set B in B(Y), the preimage f inverse (B) belongs to B(X). When Y carries the usual Borel sigma-algebra (for example Y = R with its standard topology), it is common to state that f is Borelmessbar if the preimage of every open set in Y is Borel in X.

Examples and properties: Every continuous function between topological spaces is Borelmessbar, since the preimage of any open set is open and hence Borel. More generally, any function that is measurable with respect to B(X) and B(Y) is Borelmessbar. In real analysis, a real-valued function f: R → R is Borelmessbar precisely when the preimage of every open interval is a Borel set; this is equivalent to the preimage of every open set being Borel.

Relation to other measurability concepts: The Borel sigma-algebra is the smallest sigma-algebra containing all open sets. Borel measurability is weaker than other forms of measurability such as Lebesgue measurability, which uses a larger sigma-algebra. A function can be Lebesgue measurable without being Borel measurable depending on the domain’s sigma-algebra.

Uses: Borelmessbare functions are foundational in integration, probability, and functional analysis, enabling the definition of random variables, expectations, and the study of convergence and transformation of measurable functions.