نظریه اندازه

Measure Teory

هدف اصلی نویسنده در این کتاب نمایش آن قسمت از نظریه اندازه است که در سال های قبل از انتشار کتاب تاثیر بسیار مثبت و کاربردی خود را در زمینه آنالیز مدرن به نمایش گذاشته بود. و همین موضوع سبب می شود که این کتاب هم برای دانشجویان مفید واقع شود و هم منبعی مناسب برای محققان در زمینه ریاضیات پیشرفته باشد. 

Contents

CHAPTER 1: SETS AND CLASSES

1. Set inclusion
2. Unions and intersections
3. Limits, complements, and differences
4. Ring and algebras
5. Generated rings and \(\sigma\)- rings
6. Monotone classes

CHAPTER 2: MEASURE AND OUTER MEASURES

1. Measure on rings
2. Measure on intervals
3. Properties of measures
4. Outer measures
5. Measurable sets

CHAPTER 3: EXTENSION OF MEASURES

1. Properties of induced measures
2. Extension, completion, and approximation
3. Inner measure
4. Lebesgue measure
5. Non measurable sets

CHAPTER 4: MEASURABLE FUNCTIONS

1. Measure spaces
2. Measurable functions
3. Combinations of measurable functions
4. Sequences of measurable functions
5. Pointwise convergence
6. Convergence in measure

CHAPTER 5: INTEGRATION

1. Integrable simple functions
2. Sequences of integrable simple functions
3. Integrable functions
4. Sequences of integrable functions

CHAPTER 6: GENERAL SET FUNCTIONS

1. Signed measure
2. Hahn and Jordan decompositions
3. Absolute continuity
4. The Radon-Nikodym theorem
5. Derivatives of signed measures

CHAPTER 7: PRODUCT SPACES

1. Cartesian products
2. Sections
3. Product measures
4. Fubini's theorem
5. Finite dimensional product spaces
6. Infinite dimensional product spaces

CHAPTER 8: TRANSFORMATION AND FUNCTIONS

1. Measurable transformations
2. Measure rings
3. The isomorphism theorem
4. Function spaces
5. Set functipns and point functions

CHAPTER 9: PROBABILITY

1. Heuristic introduction
2. Independence
3. Series of independent functions
4. The law of large numbers
5. Conditional probabilities and expectations
6. Measures on product spaces

CHAPTER 10: LOCALLY COMPACT SPACES

1. Topological lemmas
2. Borel sets and Baire sets
3. Regular measures 
4. Generations of Borel measures
5. Regular contents
6. Classes of continuous functions
7. Linear functionals

CHAPTER 11: HAAR MEASURE

1. Full subgroups
2. Existence
3. Measurable groups
4. Uniqueness

CHAPTER 12: MEASURE AND TOPOLOGY IN GROUPS

1. Topology in terms of measure
2. Weil topology
3. Quotient groups
4. The regularity of Haar measure

Refrences

Bibliography

List of frequently used symlobs

Index