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2. Lebesgue integral - Wikipedia

   en.wikipedia.org/wiki/Lebesgue_integral

   The Lebesgue integral describes better how and when it is possible to take limits under the integral sign (via the monotone convergence theorem and dominated convergence theorem). While the Riemann integral considers the area under a curve as made out of vertical rectangles, the Lebesgue definition considers horizontal slabs that are not ...

3. Integration by parts - Wikipedia

   en.wikipedia.org/wiki/Integration_by_parts

   This visualization also explains why integration by parts may help find the integral of an inverse function f −1 (x) when the integral of the function f(x) is known. Indeed, the functions x(y) and y(x) are inverses, and the integral ∫ x dy may be calculated as above from knowing the integral ∫ y dx.

4. Differential form - Wikipedia

   en.wikipedia.org/wiki/Differential_form

   The integral of an n-form ω on an n-dimensional manifold is defined by working in charts. Suppose first that ω is supported on a single positively oriented chart. On this chart, it may be pulled back to an n-form on an open subset of R n. Here, the form has a well-defined Riemann or Lebesgue integral as before.

5. Lebesgue integrability - Wikipedia

   en.wikipedia.org/wiki/Lebesgue_integrability

   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. Confusingly, this result is due to Lebesgue, but refers to the Riemann integral, not the Lebesgue integral.

6. Absolute continuity - Wikipedia

   en.wikipedia.org/wiki/Absolute_continuity

   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 .

7. 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 ...

8. 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

9. 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.