Ickeintegrerbara

Ickeintegrerbara (often written icke-integrerbara) is a term used in real analysis to describe functions that do not possess a Riemann integral on a given closed interval. In practice, the label highlights difficulties for Riemann integration, even though alternative notions of integration, such as the Lebesgue integral, may still apply. The concept is most commonly discussed in contrast to Riemann- and Lebesgue-integrable functions.

A central result is Lebesgue’s criterion for Riemann integrability: a bounded function on a closed interval [a, b] is Riemann integrable if and only if its set of discontinuities has Lebesgue measure zero. This explains why some highly irregular functions are not Riemann integrable. A classic example of a non-integrable function in the Riemann sense is the Dirichlet function, defined as f(x) = 1 if x is rational and f(x) = 0 if x is irrational on [0, 1]. It is discontinuous at every point, so it is not Riemann integrable. Nevertheless, it is Lebesgue integrable with integral 0 because the rationals form a set of measure zero.

Non-integrability can also arise from unboundedness on the interval, in which case the proper Riemann integral does not exist, though one may consider improper integrals or other integration theories. Hence, ickeintegrerbara often signals a distinction between Riemann integrability and other forms of integration.

See also: Riemann integral, Lebesgue integral, Dirichlet function, Lebesgue criterion.