Integrable numerical functions
Bounded Variation Functions

Riemann integration was the first rigorous approach introduced historically (1854, 1868). However, this notion proved too limited, as it led to too few integrable functions. Lebesgue integration (1904) compensated for this shortcoming, making it possible to obtain integrals of a greater number of functions.

The Riemann integral of a numerical function is obtained by partitioning its domain of definition and defining step functions, while the Lebesgue integral of a function is obtained by partitioning its domain of value and defining stepped functions (figure 1 ). Riemann's theory of integration is based on lower and upper Darboux sums, while Lebesgue's theory requires more sophisticated notions of set measures and...