Lebesgueszabály

Lebesgueszabály, or Lebesgue's criterion for Riemann integrability, is a result in real analysis that characterizes when a bounded function on a closed interval is Riemann integrable. The criterion states that a bounded function f on [a, b] is Riemann integrable if and only if the set of points where f is discontinuous has Lebesgue measure zero. In other words, the irregular points form a “small” set in the sense of Lebesgue measure.

Interpretation and consequences: The rule connects Riemann integrability with measure theory. It implies that functions with only “rare” discontinuities can be integrated in the Riemann sense. For example, Dirichlet’s function, which is discontinuous at every point of [a, b], is not Riemann integrable because the set of discontinuities has positive measure. Conversely, Thomae’s function has discontinuities only at the rational numbers (a countable set, hence measure zero) and is Riemann integrable with integral zero.

Historical context: The result is associated with Henri Lebesgue, whose development of measure theory and the Lebesgue integral in the early 20th century provided tools to precisely formulate this criterion. Lebesgue’s criterion is a foundational link between Riemann integration and measure theory, clarifying which bounded functions can be integrated in the classical Riemann sense without requiring explicit Riemann sums.

See also: Riemann integral, Lebesgue integral, Lebesgue measure, Dirichlet function, Thomae’s function.