Limitations of Lebesgue Integral
The main purpose of Lebesgue integral is to provide an integral notation where limits of integrals hold under mild assumptions. There is no guarantee that every function is Lebesgue integrable. It may happen that improper (Riemann) integral may exist for functions that are not Lebesgue integrable. One example would be . This function is not Lebesgue integrable as . On the other hand, it exists as an improper Riemann integral and the integral can be computed to be finite. An equivalent concept of improper Lebesgue integral does not exist because such a perspective is unnecessary from the viewpoint of the convergence theorems.
Read more about this topic: Lebesgue Integration
Famous quotes containing the words limitations and/or integral:
“The only rules comedy can tolerate are those of taste, and the only limitations those of libel.”
—James Thurber (1894–1961)
“Painting myself for others, I have painted my inward self with colors clearer than my original ones. I have no more made my book than my book has made me—a book consubstantial with its author, concerned with my own self, an integral part of my life; not concerned with some third-hand, extraneous purpose, like all other books.”
—Michel de Montaigne (1533–1592)