Riemann integral

The Riemann integral is the classical definition of the definite integral, named after Bernhard Riemann, in which the area under a curve is approximated by dividing the domain into intervals and summing the areas of rectangles whose heights are determined by the function's value at chosen points within each interval. As the width of the largest interval approaches zero, the sum of these rectangular areas — the Riemann sum — converges to the integral, provided the function is sufficiently well-behaved.

The Riemann integral captures the intuitive geometric notion of area and is adequate for continuous functions and most functions encountered in elementary analysis. But it fails for functions that are too irregular — functions with dense discontinuities, functions that oscillate infinitely, functions that cannot be partitioned into intervals that behave predictably. These pathological cases are not merely curiosities; they arise naturally in the limits of sequences of well-behaved functions and in the probabilistic settings that motivated the development of measure theory.

The Lebesgue integral replaces the Riemann integral by partitioning the range rather than the domain, enabling the integration of functions far more irregular than the Riemann framework can handle. The Riemann integral is not wrong; it is a special case — the case where the domain partitions cleanly and the function respects the partition. In this sense, the Riemann integral is to the Lebesgue integral what classical mechanics is to relativistic mechanics: a limiting case that works beautifully within its domain and fails catastrophically outside it.