| | PlanetMath: Comparison between Lebesgue and Riemann Integration |
 | | Functions like Sin(x)/x multiplied by the characteristic function of the set of irrational numbers would have been integrable in the first sense. |
| | The only reason that the Dirichlet function is Lebesgue, but not Riemann, integrable, is that its spikes occur on the rationals, a set of numbers which is, in comparison to the irrational numbers, a very small set. |
| | The class of Lebesgue integrable functions has the desired abstract properties (simple conditions to check whether the exchange of integral and limit is allowed), whereas the class of Riemann integrable functions does not. |
| planetmath.org /encyclopedia/ComparisonBetweenLebesgueAndRiemannIntegration2.html (1078 words) |
