Measurement and Integration
Authors
Description
This book attempts to fill the increasingly pressing need to provide, within a Master's program in Mathematics or Mathematical Statistics, a version of Measure Theory that allows for more or less immediate application to other areas such as Probability Theory, Mathematical Statistics, and their ramifications.
Target audience
Higher education
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 Set Operations
0.3 Limits and indicators
0.4 Functions
Exercises
CHAPTER 1 CLASSES OF SETS
Exercises
CHAPTER 2 MEASUREMENTS AND EXTENSION OF MEASUREMENTS
2.1 Measurements
2.2 Extension of measures
2.3 Interior measurement
2.4 Construction of measurements on semi-rings. Examples
2.5 Some comments on the measurement problem
Exercises
CHAPTER 3 MEASURABLE FUNCTIONS
Exercises
CHAPTER 4 INTEGRATION
4.1 Integral. Basic convergence theorems
4.2 Spaces L p
4.3 Applications
Exercises
CHAPTER 5 IMAGES OF MEASUREMENTS AND PRODUCT MEASUREMENTS
5.1 Measurement Images
5.2 Product measures – Fubini’s Theorem
5.3 Examples and applications
5.4 Finite and infinite products of measure spaces
Exercises
CHAPTER 6 MEASUREMENTS WITH SIGNS
6.1 Generalities
6.2 Hahn-Jordan Decomposition
6.3 Absolute continuity. Radon-Nikodym theorem
6.4 Riesz Representation Theorem
Exercises
CHAPTER 7 INTEGRATION AND DIFFERENTIATION
7.1 Differentiation of monotonic functions
7.2 Functions of bounded variation
7.3 Lebesque Differentiation Theorem
7.4 Absolute continuity
7.5 Some special results and examples
7.6 Change of variables in integrals
Exercises
REFERENCES
INDEX OF NOTATIONS
ALPHABETICAL INDEX