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2. Lebesgue integral - Wikipedia

   en.wikipedia.org/wiki/Lebesgue_integral

   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.

3. Lebesgue–Stieltjes integration - Wikipedia

   en.wikipedia.org/wiki/Lebesgue–Stieltjes...

   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

4. Absolutely integrable function - Wikipedia

   en.wikipedia.org/wiki/Absolutely_integrable_function

   In Lebesgue integration, this is exactly the requirement for any measurable function f to be considered integrable, with the integral then equaling + (), so that in fact "absolutely integrable" means the same thing as "Lebesgue integrable" for measurable functions.

5. Locally integrable function - Wikipedia

   en.wikipedia.org/wiki/Locally_integrable_function

   is not locally integrable in x = 0: it is indeed locally integrable near this point since its integral over every compact set not including it is finite. Formally speaking, 1 / x ∈ L 1 , l o c ( R ∖ 0 ) {\displaystyle 1/x\in L_{1,loc}(\mathbb {R} \setminus 0)} : [ 19 ] however, this function can be extended to a distribution on the whole R ...

6. Fatou's lemma - Wikipedia

   en.wikipedia.org/wiki/Fatou's_lemma

   In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou. Fatou's lemma can be used to prove the Fatou–Lebesgue theorem and Lebesgue's dominated convergence theorem.

7. Riemann–Lebesgue lemma - Wikipedia

   en.wikipedia.org/wiki/Riemann–Lebesgue_lemma

   However, the Riemann–Lebesgue lemma does not hold for arbitrary distributions. For example, the Dirac delta function distribution formally has a finite integral over the real line, but its Fourier transform is a constant and does not vanish at infinity.

8. Dominated convergence theorem - Wikipedia

   en.wikipedia.org/wiki/Dominated_convergence_theorem

   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 ...

9. Lebesgue integrability - Wikipedia

   en.wikipedia.org/wiki/Lebesgue_integrability

   In mathematics, Lebesgue integrability may refer to: Whether the Lebesgue integral of a function is defined; this is what is most often meant. The Lebesgue integrability condition , which determines whether the Riemann integral of a function is defined.