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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 ...
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.
The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known ... and is of use in the general theory of stochastic integration.
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 ...
The derivative of this integral at x is defined to be | |, where |B| denotes the volume (i.e., the Lebesgue measure) of a ball B centered at x, and B → x means that the diameter of B tends to 0. The Lebesgue differentiation theorem ( Lebesgue 1910 ) states that this derivative exists and is equal to f ( x ) at almost every point x ∈ R n . [ 1 ]
These Lebesgue-measurable sets form a σ-algebra, and the Lebesgue measure is defined by λ(A) = λ*(A) for any Lebesgue-measurable set A. The existence of sets that are not Lebesgue-measurable is a consequence of the set-theoretical axiom of choice , which is independent from many of the conventional systems of axioms for set theory .
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.
Equivalence between (1) and (3) is known as the fundamental theorem of Lebesgue integral calculus, due to Lebesgue. [ 3 ] For an equivalent definition in terms of measures see the section Relation between the two notions of absolute continuity .