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.
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