integrerbart

Integrerbart is a term used in Swedish mathematics to describe when a function can be assigned a finite integral under a given theory of integration. In common usage, integrerbarhet is tied to the standard notions of integrability in analysis, most often with respect to a measure.

The most widely used notion is Lebesgue integrability. A real- or complex-valued function f defined on a measure space (X, Σ, μ) is Lebesgue integrable if the integral of its absolute value is finite, that is ∫X |f| dμ < ∞. When this holds, the integral ∫X f dμ is well defined and finite. The set of all Lebesgue integrable functions on X with respect to μ is denoted L1(X, μ), and these functions form a normed space with the L1 norm ∥f∥1 = ∫X |f| dμ.

A more classical notion is Riemann integrability, applicable to functions on a finite interval [a, b]. A bounded function f is Riemann integrable if and only if the set of its discontinuities on [a, b] has Lebesgue measure zero. While every Riemann integrable function on [a, b] is Lebesgue integrable, the converse is not always true in general settings.

Examples illustrate the concept: the function f(x) = 1/√x on [0, 1] is Lebesgue integrable, with ∫0^1 x^(-1/2) dx = 2. In probability theory and functional analysis, integrable functions commonly represent expected values and inner products, and they are central to defining L1 spaces and related convergence theorems.

See also: Lebesgue integration, Riemann integration, L1 space, measure theory.