Integration

2021 ◽  
pp. 80-102
Author(s):  
James Davidson
Keyword(s):  
First Principles ◽  
Convergence Theorem ◽  
Measure Space ◽  
Product Measure ◽  
Monotone Convergence ◽  
General Measure ◽  
Fundamental Properties ◽  
Monotone Convergence Theorem ◽  
Multiple Integrals ◽  
Nikodym Theorem

The concept of an integral on a general measure space is developed from first principles. Riemann–Stieltjes and Lebesgue–Stieltjes integrals are defined. The monotone convergence theorem, fundamental properties of integrals, and related inequalities are covered. Other topics include product measure and multiple integrals, Fubini’s theorem, signed measures, and the Radon–Nikodym theorem.

scholarly journals On Fuzzy Henstock-Kurzweil-Stieltjes-♦-Double Integral on Time scales

2021 ◽  
Vol 2 (2) ◽  
pp. 38-49
Author(s):  
David AFARIOGUN ◽  
Adesanmi MOGBADEMU ◽  
Hallowed OLAOLUWA
Keyword(s):  
Uniform Convergence ◽  
Time Scales ◽  
Convergence Theorem ◽  
Monotone Convergence ◽  
Double Integral ◽  
Monotone Convergence Theorem ◽  
Integrable Functions ◽  
Dominated Convergence

We introduce and study some properties of fuzzy Henstock-Kurzweil-Stietljes-$ \Diamond $-double integral on time scales. Also, we state and prove the uniform convergence theorem, monotone convergence theorem and dominated convergence theorem for the fuzzy Henstock-Kurzweil Stieltjes-$\Diamond$-double integrable functions on time scales.

On (L1)* for General Measure Spaces

1963 ◽  
Vol 6 (2) ◽  
pp. 211-229 ◽  
Author(s):  
H. W. Ellis ◽  
D. O. Snow
Keyword(s):  
Measurable Function ◽  
Measure Space ◽  
Finite Measure ◽  
Signed Measure ◽  
General Measure ◽  
Measure Spaces ◽  
Arbitrary Measure ◽  
Image Position ◽  
Nikodym Theorem

It is well known that certain results such as the Radon-Nikodym Theorem, which are valid in totally σ -finite measure spaces, do not extend to measure spaces in which μ is not totally σ -finite. (See §2 for notation.) Given an arbitrary measure space (X, S, μ) and a signed measure ν on (X, S), then if ν ≪ μ for X, ν ≪ μ when restricted to any e ∊ Sf and the classical finite Radon-Nikodym theorem produces a measurable function ge(x), vanishing outside e, with

scholarly journals Extended Real-Valued Double Sequence and Its Convergence

2015 ◽  
Vol 23 (3) ◽  
pp. 253-277 ◽  
Author(s):  
Noboru Endou
Keyword(s):  
Convergence Theorem ◽  
Double Sequence ◽  
Monotone Convergence ◽  
Double Sequences ◽  
Monotone Convergence Theorem ◽  
Fatou’S Lemma ◽  
Fatou's Lemma

Abstract In this article we introduce the convergence of extended realvalued double sequences [16], [17]. It is similar to our previous articles [15], [10]. In addition, we also prove Fatou’s lemma and the monotone convergence theorem for double sequences.

scholarly journals The Lebesgue Monotone Convergence Theorem

2008 ◽  
Vol 16 (2) ◽  
Author(s):  
Noboru Endou ◽  
Keiko Narita ◽  
Yasunari Shidama
Keyword(s):  
Convergence Theorem ◽  
Monotone Convergence ◽  
Monotone Convergence Theorem

A new proof of the monotone convergence theorem of Lebesgue integral on σ-class

2012 ◽  
Vol 62 (6) ◽  
Author(s):  
Dinh Hoa
Keyword(s):  
Convergence Theorem ◽  
Short Note ◽  
Monotone Convergence ◽  
Lebesgue Integral ◽  
Monotone Convergence Theorem

AbstractIn this short note a new proof of the monotone convergence theorem of Lebesgue integral on σ-class is given.

Sign in / Sign up

Export Citation Format

Share Document