Borelmålebar

Borelmålebar, in Swedish mathematical usage, refers to the property of a function being measurable with respect to the Borel sigma-algebra. Specifically, let X be a set equipped with a sigma-algebra F, and let Y be a topological space equipped with its Borel sigma-algebra B. A function f: X → Y is borelmålebar if for every Borel set B in Y, the preimage f^{-1}(B) belongs to F. When Y = R with the standard Borel sigma-algebra, this condition is often stated as: the preimage of every open set in R is measurable, or equivalently for every real a, the set {x ∈ X : f(x) > a} is in F.

In practical terms, borelmålebar functions are those whose behavior can be described in terms of the topology of the target space. For real-valued functions on metric spaces, several equivalent criteria hold: preimages of open sets are measurable, or preimages of half-lines (−∞, a) are measurable for all a. Continuous functions are borelmålebar, and, more generally, many functions constructed from simple, well-understood pieces are borelmålebar.

Relation to Lebesgue measurability: every borelmålebar function is Lebesgue-measurable, but the converse is not always true. The Borel sigma-algebra is smaller than the Lebesgue sigma-algebra, so some Lebesgue-measurable functions need not be borelmålebar. In probability theory, borelmålebar functions commonly serve as random variables when defined on a probability space with the appropriate sigma-algebras.

Historically, the concept derives from the Borel sigma-algebra introduced by Émile Borel in the development of measure theory; the term borelmålebar is the Swedish terminology for Borel-measurable.