The Lebesgue-Nikodym Theorem for Vector Valued Radon Measures

1974 ◽  
Vol 0 (139) ◽  
pp. 0-0 ◽  
Author(s):  
Erik Thomas
Keyword(s):  
Radon Measures ◽  
Vector Valued ◽  
Nikodym Theorem

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.

Topological measure spaces: two counter-examples

1975 ◽  
Vol 78 (1) ◽  
pp. 95-106 ◽  
Author(s):  
D. H. Fremlin
Keyword(s):  
Radon Measures ◽  
Topological Measure ◽  
Measure Spaces ◽  
Finite Measures ◽  
Nikodym Theorem

The ‘Radon measures’ of N. Bourbaki(1) enjoy many striking properties. Among the most important of these is the ‘strong Radon-Nikodým theorem’ that the dual of L1-(μ) can always be identified with L∞(μ) ((1), chap. 5, §5, no. 8, theorem 4). As this is certainly not true of non-σ-finite measures in general, it is natural to ask what are the special properties on which it relies.

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