Integration Theory

2021 ◽  
pp. 341-444
Author(s):  
Adel N. Boules
Keyword(s):  
Linear Functional ◽  
Integration Theory ◽  
General Measure ◽  
Radon Measures ◽  
Lp Spaces ◽  
Approximation Theorems ◽  
Positive Linear Functional ◽  
Product Measures ◽  
Compactly Supported Functions ◽  
Nikodym Theorem

The Lebesgue measure on ?n (presented in section 8.4) is a pivotal component of this chapter. The approach in the chapter is to extend the positive linear functional provided by the Riemann integral on the space of continuous, compactly supported functions on ?n (presented in section 8.1). An excursion on Radon measures is included at the end of section 8.4. The rest of the sections are largely independent of sections 8.1 and 8.4 and constitute a deep introduction to general measure and integration theories. Topics include measurable spaces and measurable functions, Carathéodory’s theorem, abstract integration and convergence theorems, complex measures and the Radon-Nikodym theorem, Lp spaces, product measures and Fubini’s theorem, and a good collection of approximation theorems. The closing section of the book provides a glimpse of Fourier analysis and gives a nice conclusion to the discussion of Fourier series and orthogonal polynomials started in section 4.10.

Are Non-Commutative Lp Spaces Really Non-Commutative?

1981 ◽  
Vol 33 (6) ◽  
pp. 1319-1327 ◽  
Author(s):  
A. Katavolos
Keyword(s):  
Measure Space ◽  
Linear Functional ◽  
Von Neumann Algebra ◽  
Hölder Inequality ◽  
Integration Theory ◽  
Von Neumann ◽  
Lp Spaces ◽  
Positive Linear Functional ◽  
Image Position ◽  
Abelian Von Neumann Algebra

1. The central objects in integration theory can be considered to be an abelian Von Neumann algebra, L∞, of the measure space, together with a (not necessarily finite-valued) positive linear functional on it, the integral (see [10]). It is natural, therefore, to attempt to construct a “non-commutative” integration theory starting with a non-abelian Von Neumann algebra. Segal [9] and Dixmier [2] have developed such a theory, and constructed the Non-Commutative Lp spaces associated with a Von Neumann algebra M and a normal, faithful, semifinite trace (i.e. a unitarily invariant weight) t on M. They show that there exists a unique ultra-weakly dense *-ideal J of M such that t (extends to) a positive linear form on J . A generalisation of the Hölder inequality then shows that, for 1 ≦ p < ∞, the functionis a norm on J, denoted by || • ||p.

Commutative L-algebras and measure theory

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wolfgang Rump
Keyword(s):  
Measure Theory ◽  
Measurable Space ◽  
Universal Covering ◽  
Continuous Functions ◽  
Stone Space ◽  
Integration Theory ◽  
General Measure ◽  
Lattice Order ◽  
Finitely Additive Measures ◽  
Additive Measures

Abstract Measure and integration theory for finitely additive measures, including vector-valued measures, is shown to be essentially covered by a class of commutative L-algebras, called measurable algebras. The domain and range of any measure is a commutative L-algebra. Each measurable algebra embeds into its structure group, an abelian group with a compatible lattice order, and each (general) measure extends uniquely to a monotone group homomorphism between the structure groups. On the other hand, any measurable algebra X is shown to be the range of an essentially unique measure on a measurable space, which plays the role of a universal covering. Accordingly, we exhibit a fundamental group of X, with stably closed subgroups corresponding to a special class of measures with X as target. All structure groups of measurable algebras arising in a classical context are archimedean. Therefore, they admit a natural embedding into a group of extended real-valued continuous functions on an extremally disconnected compact space, the Stone space of the measurable algebra. Extending Loomis’ integration theory for finitely additive measures, it is proved that, modulo null functions, each integrable function can be represented by a unique continuous function on the Stone space.

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

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