دانلود کتاب Measure theory and integration

- approaches integration via measure theory, as opposed to measure theory via integration, making it easier to understand the subject
- includes numerous worked examples necessary for teaching and learning at undergraduate level
- detailed solutions are provided for the 300 problem exercises which test comprehension of the theorems provided

This text approaches integration via measure theory as opposed to measure theory via integration, an approach which makes it easier to grasp the subject. Apart from its central importance to pure mathematics, the material is also relevant to applied mathematics and probability, with proof of the mathematics set out clearly and in considerable detail. Numerous worked examples necessary for teaching and learning at undergraduate level constitute a strong feature of the book, and after studying statements of results of the theorems, students should be able to attempt the 300 problem exercises which test comprehension and for which detailed solutions are provided.

Contents

Preliminaries
- Set theory
- Topological ideas
- Sequences and limits
- Functions and mappings
- Cardinal numbers and countability
- Further properties of open sets
- Cantor-like sets

Measure on the real line
- Lebesgue Outer measure
- Measurable sets
- Regularity
- Measurable functions
- Borel and Lebesgue measurability
- Hausdorff measures on the real line

Integration of functions of a real variable
- Integration of non-negative functions
- The general integral
- Integration of series
- Riemann and Lebesgue integrals

Differentiation
- The four derivates
- Lebesgue’s differentiation theorem
- Differentiation and integration
- The Lebesgue set

Abstract measure spaces
- Abstract measure spaces
- Measures and outer measures
- Extension of a measure
- Uniqueness of the extension
- Completion of a measure
- Measure spaces
- Integration with respect to a measure

Inequalities and the Lp spaces
- TheLpSpaces
- Convex functions
- Jensen’s inequality
- The inequalities of Holder and Minkowski
- Completeness of Lp(u)

Convergence
- Convergence in measure
- Almost uniform convergence
- Convergence diagrams
- Counter examples

Signed measures and their derivatives
- Signed measures and the Hahn decomposition
- The Jordan decomposition
- The Radon-Nikodym theorem
- Some applications of the Radon-Nikodym theorem
- Bounded linear functionals on Lp

Lebesgue-stieljes integration
- Lebesgue-Stieltjes measure
- Applications to Hausdorff measures
- Absolutely continuous functions
- Integration by parts
- Change of variable
- Riesz representation theorem for C(I)

Measure and integration in a product space
- Measuring in a product space
- The product measure and Fubini’s theorem
- Lebesgue measure in Euclidean space
- Laplace and Fourier transforms

Hints and answers to exercises

References