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Furthermore, the Lebesgue integral can be generalized in a straightforward way to more general spaces, measure spaces, such as those that arise in probability theory. The term Lebesgue integration can mean either the general theory of integration of a function with respect to a general measure, as introduced by Lebesgue, or the specific case of ...
4 Measure theory and the Lebesgue integral. 5 Extensions. 6 Integral equations. ... This is a list of integration and measure theory topics, by Wikipedia page.
Henri Léon Lebesgue ForMemRS [1] (/ l ə ˈ b ɛ ɡ /; [3] French: [ɑ̃ʁi leɔ̃ ləbɛɡ]; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of a function defined for that axis.
Lebesgue's dominated convergence theorem is a special case of the Fatou–Lebesgue theorem. Below, however, is a direct proof that uses Fatou’s lemma as the essential tool. Since f is the pointwise limit of the sequence ( f n ) of measurable functions that are dominated by g , it is also measurable and dominated by g , hence it is integrable.
An alternative approach (Hewitt & Stromberg 1965) is to define the Lebesgue–Stieltjes integral as the Daniell integral that extends the usual Riemann–Stieltjes integral. Let g be a non-decreasing right-continuous function on [ a , b ] , and define I ( f ) to be the Riemann–Stieltjes integral
The stronger versions of Fubini's theorem on a product of two unit intervals with Lebesgue measure, where the function is no longer assumed to be measurable but merely that the two iterated integrals are well defined and exist, are independent of the standard Zermelo–Fraenkel axioms of set theory.
In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non-negative pointwise-increasing measurable functions (), taking the integral and the supremum can be interchanged with the result being finite if either one is ...
In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue greatly improved measure theory, and introduced his own theory of integration, now known as Lebesgue integration, which proved to be a big improvement over Riemann's. Hilbert introduced Hilbert spaces to solve integral equations.