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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. 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. 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.

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. Null set - Wikipedia

   en.wikipedia.org/wiki/Null_set

   Null sets play a key role in the definition of the Lebesgue integral: if functions and are equal except on a null set, then is integrable if and only if is, and their integrals are equal. This motivates the formal definition of L p {\displaystyle L^{p}} spaces as sets of equivalence classes of functions which differ only on null sets.

7. 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 .

8. 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 ...

9. Integration by substitution - Wikipedia

   en.wikipedia.org/wiki/Integration_by_substitution

   For Lebesgue measurable functions, the theorem can be stated in the following form: [6] Theorem — Let U be a measurable subset of R n and φ : U → R n an injective function , and suppose for every x in U there exists φ ′( x ) in R n , n such that φ ( y ) = φ ( x ) + φ′ ( x )( y − x ) + o (‖ y − x ‖) as y → x (here o is ...