A theory of integration and measure which greatly extended the Riemann integral. Riemann integrable functions are bounded and defined on bounded intervals, whereas Lebesgue’s theory was much more general and also included powerful convergence theorems (see dominated convergence theorem, monotone convergence theorem). The existence of a set or function which is not Lebesgue measurable requires axioms beyond the Zermelo-Fraenkel axioms, such as the axiom of choice (see vitali set).