Integrerbare

Integrerbare is the Danish term for integrable, used to describe functions or quantities for which an integral can be assigned in a well-defined sense. In mathematics, integrability is defined relative to a chosen theory of integration, most commonly Riemann or Lebesgue integration.

Riemann integrable: A bounded function f on a closed interval [a, b] is Riemann integrable if the limit of Riemann sums exists as the partition norm tends to zero. A standard criterion is that the set of points where f is discontinuous has Lebesgue measure zero. When these conditions are met, the Riemann integral exists and equals the common limit of the upper and lower sums.

Lebesgue integrable: A measurable function f on a measure space is Lebesgue integrable if the integral of its absolute value is finite, i.e., ∫ |f| < ∞. Such functions are said to be integrable in the Lebesgue sense, and their integral ∫ f is defined. Lebesgue integration extends the concept to a broader class of functions and is central to modern analysis and probability theory.

Examples: All continuous functions on a closed interval are Riemann integrable. Step functions and many bounded functions with mild discontinuities are also Riemann integrable. The function that is 1 on the rationals and 0 on the irrationals within [0, 1] is not Riemann integrable, but it is Lebesgue integrable with integral 0.

Relation to other concepts: Integrable functions form spaces such as L1, consisting of functions whose absolute value has finite integral. In physics and statistics, integrability underlies calculations of areas, probabilities, expectations, and various physical quantities.

Origin and usage: The concept reflects the ability to assign a numerical value to accumulation or area through a defined process of integration, with Riemann developing the approach in the 19th century and Lebesgue expanding it in the early 20th century.