Borelfunktion

A BorelFunktion, or Borel function, is a function that is measurable with respect to the Borel sigma-algebra on the real line. More precisely, if f maps a topological space X into the real numbers R, then f is Borel measurable if for every open set U ⊂ R, the preimage f^{-1}(U) is a Borel set in X. In the common real-valued setting on R with its standard topology, this means f^{-1}(U) belongs to the Borel sigma-algebra B(R) for every open U.

The Borel sigma-algebra B(R) is the smallest sigma-algebra containing all open subsets of R. It also contains all intervals, closed sets, and many more sets obtained through countable unions, intersections, and complements. Borel sets form a natural class for defining measurability before invoking more general Lebesgue measurability.

Every continuous function f: X → R is Borel measurable, since preimages of open sets under a continuous map are open (hence Borel). The class of Borel functions is broad: it is closed under standard algebraic operations (sum, difference, product, and division where defined) and under pointwise limits of sequences of Borel functions. Indicator functions of Borel sets are Borel measurable, as are many simple and step functions.

A key relation is that every Borel measurable function is Lebesgue measurable, but not every Lebesgue measurable function is Borel measurable. In probability and analysis, random variables are often modeled as Borel measurable functions with respect to the chosen underlying sigma-algebra.

Notes: the term BorelFunktion reflects the German usage; in English contexts the standard term is Borel measurable function or simply Borel function.