Riemann Integration — Formal Definition

GRE Math Subject · Axiom Academy

LESSON Riemann Integration — Formal Definition Partitions, Darboux sums, integrability criteria, and the Fundamental Theorem of Calculus Step 1: Partitions and Darboux Sums Definition: A partition of is a finite set with . Given a bounded function and a partition , define on each subinterval : Refinement lemma: If is a refinement of (i.e., ), then Step 2: The Riemann (Darboux) Integral A bounded function is Riemann integrable on if and only if: When they agree, their common value is . Riemann's Criterion: is Riemann integrable on if and only if for every , there exists a partition such that: In words: the upper and lower sums can be made arbitrarily close by choosing a fine enough partition. Step 3: Which Functions Are Integrable? Lebesgue's Criterion: A bounded function is Riemann integrable if and only if its set of discontinuities has Lebesgue measure zero . This powerful result immediately tells us: Continuous functions are integrable (zero discontinuities) Monotone functions on are integrable (at most countably many discontinuities — measure zero) Functions with finitely many discontinuities are integrable Piecewise continuous functions are integrable Classic Non-Integrable Function: The Dirichlet function is not Riemann integrable. Its set of discontinuities is all of (measure > 0). For every partition, and .

This is the written version of the interactive lesson above. See the full GRE Math Subject course.