13 Facts About Lebesgue integral

The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions.

Long before the 20th century, mathematicians already understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the area under the curve could be defined as the Lebesgue integral, and computed using approximation techniques on the region by polygons.

Lebesgue integral plays an important role in probability theory, real analysis, and many other fields in mathematics.

The Riemann Lebesgue integral—proposed by Bernhard Riemann —is a broadly successful attempt to provide such a foundation.

The Lebesgue integral is better able to describe how and when it is possible to take limits under the integral sign .

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Lebesgue integral summarized his approach to integration in a letter to Paul Montel:.

An equivalent way to introduce the Lebesgue integral is to use so-called simple functions, which generalize the step functions of Riemann integration.

Theory of the Lebesgue integral requires a theory of measurable sets and measures on these sets, as well as a theory of measurable functions and integrals on these functions.

One approach to constructing the Lebesgue integral is to make use of so-called simple functions: finite, real linear combinations of indicator functions.

The Riemann Lebesgue integral is inextricably linked to the order structure of the real line.

The Riemann Lebesgue integral exists for any continuous function of compact support defined on .

Furthermore, the Riemann Lebesgue integral is a uniformly continuous functional with respect to the norm on, which is dense in.

Main purpose of the Lebesgue integral is to provide an integral notion where limits of integrals hold under mild assumptions.