Theory of Measurement Course
Authors
Description
This textbook is aimed at those wishing to study Analysis and related subjects in depth, with a view to studying for a doctorate. In the book there is a gradation in the hypotheses underlying the theorems and in this way it is hoped to convey to students the exact dimension of the generality with which certain concepts are validated, as well as obtaining results with economic proofs and whose hypotheses are easy to remember.
Several new examples and exercises have been added to this edition. The newly proposed exercises both complement and provide an even deeper insight into the theory or points that could not be covered in the text.
This book has two new sections, one on product measures, in the very first chapter, and the other on weak topology, in Chapter 10, which deals with Borelian measures. The latter section is due to its relationship with the Riesz-Markov Theorem and its importance for research in Ergodic Analysis and Theory. In general, this relevant topic is banished to limbo by the various disciplines that should deal with it in detail (Measurement, Ergodic Theory and, especially, Functional Analysis).
Target audience
Higher education
Name: Theory of Measurement Course
Author(s): e Augusto Armando de Castro Júnior
Pages: 193
Publication: IMPA, 2015
ISBN: 978-85-244-0394-1
Edition: 3
Introduction
1 Measures in semi-rings and rings
1.1 Positive measures
1.2 Product measures
1.3 Regular measures
1.4 σ-additive measures
1.5 Exercises
2 Extension of measures
2.1 Simple functions
2.2 Higher integral
2.3 The extension theorem for measures
2.4 Exercises
3 The Lebesque Convergence Theorems
3.1 Measurable Functions
3.2 Monotone Convergence
3.3 Dominated Convergence
3.4 Exercises
4 Indefinite Integrals, Signed and Complex Measures
4.1 Indefinite Integrals
4.2 Signed Measures
4.3 Complex Measures
4.3 The Radon-Nikodym Theorem
4.5 Exercises
5 The Lebesque Decomposition Theorem
5.1 The space M (A)
5.2 The subspaces Mα (μ) and Ms (μ)
5.3 Exercises
6 Lp Spaces
6.1 Density of simple functions in Lp, 1 ≤ P < +∞
6.2 Comments on the non-density of simple functions
6.3 Duality between Lp (μ) and Lq (μ), 1/ p + 1/ q = 1
6.4 Exercises
7 Convergence of sequences of functions
7.1 Almost uniform convergence
7.2 Convergence in Measure
7.3 Convergence table
7.4 Exercises
8 Product measures
8.1 σ-additive and σ-algebraic classes
8.2 The Tonelli-Cavalieri theorem
8.3 The Fubini theorems
8.4 Exercises
9 Transport of measures and invariant measures
9.1 Measurable applications and image measures
9.2 Poincaré’s recurrence theorem
9.3 Exercises
10 Borelian measures in locally compact spaces
10.1 Density of continuous functions
10.2 The Riesz-Markov representation theorem
10.3 The weak-* topology in(C(X))*
10.4 Exercises
11 Derivation and Integration
11.1 Derivatives of measures
11.2 The Lebesque derivation theorem
11.3 The Fundamental Theorem of Calculus
11.4 Exercises
Bibliography
Index