Integral Theory

Checkpoints

• The Lebesgue Integral Essential The Lebesgue integral extends the Riemann integral by partitioning the range of a function instead of its domain, making it possible to integrate over arbitrary measurable sets and to interchange limits with integrals under mild conditions. This checkpoint constructs the integral in three stages — for simple functions, non-negative measurable functions, and general measurable functions — and contrasts it with the Riemann integral.