Ikkeintegrerbart

Ikkeintegrerbart is a term used in mathematics to describe a function or quantity that does not possess a finite integral under a given framework for integration. The meaning depends on which notion of integral is being used, most commonly the Riemann integral or the Lebesgue integral, and it can also refer to improper or conditionally defined integrals.

In the Riemann sense, a bounded function on a closed interval is integrable if the upper and lower sums converge to the same value. Some functions fail this criterion and are therefore ikkeintegrerbart in the Riemann sense. A classic example is the Dirichlet function, which takes the value 1 on rationals and 0 on irrationals on an interval; it is not Riemann integrable, though it is Lebesgue measurable.

In the Lebesgue sense, a function is integrable if the integral of its absolute value is finite, i.e., the function belongs to L1. A function can be Ikkeintegrerbar in Lebesgue sense if ∫|f| diverges or if the function is not measurable. Common examples include f(x)=1/x on (0,1], where the improper integral diverges, and hence the function is not Lebesgue integrable over that interval.

Distinct from non-integrability is the notion of divergent improper integrals, where an improper integral may exist in a conditional or extended sense but not yield a finite Lebesgue integral. In summary, ikkeintegrerbart signals the absence of a finite value under a specified integral framework, with variations across Riemann, Lebesgue, and improper integral contexts.

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