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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. 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 ...

4. 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.

5. 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

6. Absolutely integrable function - Wikipedia

   en.wikipedia.org/wiki/Absolutely_integrable_function

   Then | | + + + + + | | so | | + + + + + | | This shows that the sum of the four integrals (in the middle) is finite if and only if the integral of the absolute value is finite, and the function is Lebesgue integrable only if all the four integrals are finite. So having a finite integral of the absolute value is equivalent to the conditions for ...

7. 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.

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   www.aol.com/finance/watch-mlb-opening-day-free...

   Hulu with Live TV. The free trial on this service is no longer offered. It will now cost you $77 per month. YouTubeTV. After up to a two-week trial, you can expect monthly charges of $73.

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 ...