Measurement and Integration
Authors
Description
This book is an attempt to fill the increasingly urgent need to provide, in a Master’s program in Mathematics or Mathematical Statistics, a version of Measure Theory that allows a more or less immediate application to other areas such as Probability Theory, Mathematical Statistics and their ramifications.
Target audience
Higher education
Name: Measurement and Integration
Author(s): e Pedro Jesus Fernandez
Pages: 198
Publication: IMPA, 2015
ISBN: 978-85-244-0105-3
Edition: 2
PREFACE
INTRODUCTION
CHAPTER 0 SETS
0.1 Sets
0.2 Operations with sets
0.3 Limits and indicators
0.4 Functions
Exercises
CHAPTER 1 CLASSES OF SETS
Exercises
CHAPTER 2 MEASURES AND EXTENSION OF MEASURES
2.1 Measures
2.2 Extension of measures
2.3 Interior measure
2.4 Construction of measures on semi-rings. Examples
2.5 Some comments on the measurement problem
Exercises
CHAPTER 3 MESURABLE FUNCTIONS
Exercises
CHAPTER 4 INTEGRATION
4.1 Integra. Basic convergence theorems
4.2 Lp spaces
4.3 Applications
Exercises
CHAPTER 5 IMAGES OF MEASURES AND MEASURE-PRODUCT
5.1 Images of measures
5.2 Measure-product – Fubini’s theorem
5.3 Examples and applications
5.4 Finite and infinite products of measure spaces
Exercises
CHAPTER 6 SIGNAL MEASUREMENTS
6.1 General
6.2 Hahn and Jordan decomposition
6.3 Absolute continuity. Radon-Nikodym theorem
6.4 Riesz representation theorem
Exercises
CHAPTER 7 INTEGRATION AND DIFFERENTIATION
7.1 Differentiation of monotone functions
7.2 Functions of limited variation
7.3 Lebesque differentiation theorem
7.4 Absolute continuity
7.5 Some results and special examples
7.6 Change of variables in integrals
Exercises
REFERENCES
INDEX OF NOTATIONS
ALPHABETICAL INDEX