Lebeslekriteriet

Lebeslekriteriet, often translated as the Lebesque criterion for Riemann integrability, is a fundamental result in real analysis that provides a precise condition for a bounded function on a closed interval to be Riemann integrable. The criterion states that a bounded function f defined on a closed interval [a, b] is Riemann integrable if and only if the set of its discontinuities has Lebesgue measure zero.

The concept of measure zero is central to this criterion. A set has Lebesgue measure zero if it can be covered by a countable collection of intervals whose total length is arbitrarily small. For instance, the set of rational numbers within any interval has measure zero because it is countable.

Therefore, Lebeslekriteriet implies that if a function has only a "small" set of discontinuities, it will be Riemann integrable. Functions that are continuous everywhere or have a finite number of jump discontinuities on an interval are Riemann integrable, as their sets of discontinuities have measure zero. Conversely, a function with an uncountable number of discontinuities spread throughout the interval, such as the Dirichlet function (which is 1 for rational numbers and 0 for irrational numbers), will not be Riemann integrable because its set of discontinuities, the entire interval, does not have measure zero. This criterion is a powerful tool for understanding the integrability of a wide range of functions.