12-4-2014 9:30 AM
A Short Journey Through the Riemann Integral
An introductory-level theory of integration was studied, focusing primarily on the well-known Riemann integral and ending with the Lebesgue integral. An examination of the Riemann integral's basic properties and necessary conditions shows that this integral is not very strong. This conclusion leads to Lebesgue's necessary and sufficient condition for Riemann integrability, perhaps where the Riemann integral ends and the Lebesgue integral begins: a function defined on a closed integral [a, b] is Riemann integrable if and only if the function is discontinuous on a set of measure zero. The proof of this uses that the infinite union of sets of measure zero has a measure of zero. To distinguish between the Riemann and Lebesque integral, the classical example of the dirac-delta function displays the strength of the Lebesgue integral over the Riemann integral. In conclusion, every function that is Riemann integrable is Lebesgue integrable, but the converse is not true.