Measure and Integration Theory

Description

This book gives a straightforward introduction to the field as it is nowadays required in many branches of analysis and especially in probability theory. The first three chapters (Measure Theory, Integration Theory, Product Measures) basically follow the clear and approved exposition given in the author's earlier book on "Probability Theory and Measure Theory". Special emphasis is laid on a complete discussion of the transformation of measures and integration with respect to the product measure, convergence theorems, parameter depending integrals, as well as the Radon-Nikodym theorem. The final chapter, essentially new and written in a clear and concise style, deals with the theory of Radon measures on Polish or locally compact spaces. With the main results being Luzin's theorem, the Riesz representation theorem, the Portmanteau theorem, and a characterization of locally compact spaces which are Polish, this chapter is a true invitation to study topological measure theory. The text addresses graduate students, who wish to learn the fundamentals in measure and integration theory as needed in modern analysis and probability theory. It will also be an important source for anyone teaching such a course. Preface Introduction Notations Chapter I Measure Theory § 1. σ-algebras and their generators § 2. Dynkin systems § 3. Contents, premeasures, measures § 4. Lebesgue premeasure § 5. Extension of a premeasure to a measure § 6. Lebesgue-Borel measure and measures on the number line § 7. Measurable mappings and image measures § 8. Mapping properties of the Lebesgue-Borel measure Chapter II Integration Theory § 9. Measurable numerical functions § 10. Elementary functions and their integral §11. The integral of non-negative measurable functions § 12. Integrability § 13. Almost everywhere prevailing properties § 14. The spaces ℒp(μ) § 15. Convergence theorems § 16. Applications of the convergence theorems § 17. Measures with densities: the Radon-Nikodym theorem § 18.* Signed measures § 19. Integration with respect to an image measure § 20. Stochastic convergence § 21. Equi-integrability Chapter III Product Measures § 22. Products of σ-algebras and measures § 23. Product measures and Fubini’s theorem § 24. Convolution of finite Borel measures Chapter IV Measures on Topological Spaces § 25. Borel sets, Borel and Radon measures § 26. Radon measures on Polish spaces § 27. Properties of locally compact spaces § 28. Construction of Radon measures on locally compact spaces § 29. Riesz representation theorem § 30. Convergence of Radon measures § 31. Vague compactness and metrizability questions Bibliography Symbol Index Name Index Subject Index