Lebesgue integral

Noun

Meaning

Lebesgue integral (plural Lebesgue integrals)

English Wikipedia has an article on:Lebesgue integrationWikipedia
(mathematical analysis, singular only, definite and countable) An integral which has more general application than that of the Riemann integral, because it allows the region of integration to be partitioned into not just intervals but any measurable sets for which the function to be integrated has a sufficiently narrow range. (Formal definitions can be found at PlanetMath).
The Lebesgue integral is learned in a first-year real-analysis course.
Compute the Lebesgue integral of f over E.

Examples

According to Steinhaus, while he was strolling through the gardens he was surprised to overhear the term "Lebesgue integral" ( Lebesgue integration was at the time still a fairly new idea in mathematics) and walked over to investigate. Source: Internet

For a suitable class of functions (the measurable functions ) this defines the Lebesgue integral. Source: Internet

In fact, not only does this function not have an improper Riemann integral, its Lebesgue integral is also undefined (it equals ). Source: Internet

Lebesgue integration has the property that every function defined over a bounded interval with a Riemann integral also has a Lebesgue integral, and for those functions the two integrals agree. Source: Internet

If the delta function is already understood as a measure, then the Lebesgue integral of a test function against that measure supplies the necessary integral. Source: Internet

Lebesgue integral main Riemann–Darboux's integration (top) and Lebesgue integration (bottom) It is often of interest, both in theory and applications, to be able to pass to the limit under the integral. Source: Internet

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