**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:

“Growing up means letting go of the dearest megalomaniacal dreams of our childhood. Growing up means knowing they can’t be fulfilled. Growing up means gaining the wisdom and skills to get what we want within the *limitations* imposed by reality—a reality which consists of diminished powers, restricted freedoms and, with the people we love, imperfect connections.”

—Judith Viorst (20th century)

“Self-centeredness is a natural outgrowth of one of the toddler’s major concerns: What is me and what is mine...? This is why most toddlers are incapable of sharing ... to a toddler, what’s his is what he can get his hands on.... When something is taken away from him, he feels as though a piece of him—an *integral* piece—is being torn from him.”

—Lawrence Balter (20th century)