Abstract
For a continuous function on a closed bounded interval and for a function with no worse discontinuities than a few finite jumps, the Lebesgue integral is equal to the Riemann integral. For a positive unbounded function on a bounded interval and for a positive function on an unbounded interval, the Lebesgue integral has the same value as the improper Riemann integral whenever the latter exists. When it comes to ease of manipulation, the Lebesgue integral has great advantages over that of Riemann. The class of Lebesgue integrable functions is larger than the class of Riemann integrable functions, and is sufficiently large to make the rules of operation much more permissive. In the Lebesgue theory there are simple general conditions that allow passage to the limit under the integral sign, term by term integration of infinite series, inversion of the order of integration in repeated integrals, and recovery of a function from its derivative by integration.
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