Mättbara

Mättbara (measurable) is a term used in Swedish mathematics to describe objects that are measurable with respect to a given measure. In measure theory, a set A ⊆ X is mättbar with respect to a measure μ on a measurable space (X, Σ) if A belongs to the σ-algebra Σ. A function f: X → ℝ is mättbar if the preimage f−1(B) lies in Σ for every Borel set B ⊆ ℝ. When μ is the Lebesgue measure on ℝ and Σ is the Lebesgue σ-algebra, one speaks of Lebesgue mättbara sets and functions.

Examples include: all Borel sets on ℝ are mättbara with respect to the Borel σ-algebra; the indicator function 1_A is mättbar for any mättbar A; simple functions, finite sums ∑ c_i 1_{A_i} with mättbara A_i, are mättbara; the composition of a mättbar function with a Borel-measurable function is mättbar.

Key properties: The collection of mättbara sets forms a σ-algebra, closed under complementation, countable unions and intersections. The class of mättbara functions is closed under algebraic operations such as addition and multiplication, and under pointwise limits of sequences of mättbara functions under standard conditions; this underpins the rigorous development of integration and probability theory.

Etymology and usage: The term derives from Swedish mäta (to measure). In Swedish mathematical literature, mättbar is used to denote what English texts refer to as measurable. The plural adjective form mättbara describes sets or functions that are measurable with respect to a given measure.