målebarhet

Målebarhet describes whether a set or a function is compatible with a given measure in mathematics. In a measure space (X, F, μ), a subset E ⊆ X is measurable if E ∈ F. A function f: X → ℝ is measurable if the preimage of every Borel set is in F; equivalently, for every real α, the set {x ∈ X : f(x) > α} belongs to F. These definitions extend to more general target spaces with appropriate σ-algebras.

Common examples include Borel measurability on the real line, where the σ-algebra is generated by open intervals, and Lebesgue measurability, which uses the Lebesgue σ-algebra. All Borel sets are Lebesgue measurable, but the Lebesgue σ-algebra contains more sets; there exist Lebesgue measurable sets that are not Borel.

Non-measurable sets exist under widely accepted foundations: with the Axiom of Choice, there are subsets of the real numbers that are not measurable with respect to Lebesgue measure, such as certain Vitali sets.

Measurability is central to many areas of analysis and probability. It enables the definition of integrals (Lebesgue integration) and ensures that random variables are well-defined as measurable functions. The collection of measurable sets and functions is closed under standard constructions, including countable unions and intersections, complements, and limits of sequences of measurable entities.

Outside pure mathematics, målebarhet can colloquially refer to whether a property can be measured or observed in practice, but in a mathematical context it denotes compatibility with a chosen measure.