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The Lebesgue integral, named after French mathematician Henri Lebesgue, is one way to make this concept rigorous and to extend it to more general functions. The Lebesgue integral is more general than the Riemann integral, which it largely replaced in mathematical analysis since the first half of the 20th century. It can accommodate functions ...
4 Measure theory and the Lebesgue integral. 5 Extensions. 6 Integral ... This is a list of integration and measure theory topics, by Wikipedia page. Intuitive foundations
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.
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 ...
A Lebesgue-measurable set can be "squeezed" between a containing G δ set and a contained F σ. I.e, if A is Lebesgue-measurable then there exist a G δ set G and an F σ F such that G ⊇ A ⊇ F and λ(G \ A) = λ(A \ F) = 0. Lebesgue measure is both locally finite and inner regular, and so it is a Radon measure.
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 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 ]