Measure, integration and function spaces

1. Introduction. 1.1. Preliminaries. 1.2. Extended Real Numbers (R*) and R[superscript n]. 1.3. Lebesgue's Definition of the Integral
2. Measure Theory. 2.1. Semi-rings and Algebras of Sets. 2.2. Additive Set Functions. 2.3. Outer Measures. 2.4. Extensions of Premeasures. 2.5. Lebesgue Measure. 2.6. Lebesgue-Stieltjes Measure. 2.7. Regular Measures. 2.8. The Nikodym Convergence and Boundedness Theorems
3. Integration. 3.1. Measurable Functions. 3.2. The Lebesgue Integral. 3.3. The Riemann and Lebesgue Integrals. 3.4. Integrals Depending on a Parameter. 3.5. Convergence in Mean. 3.6. Convergence in Measure. 3.7. Comparison of Modes of Convergence. 3.8. Mikusinski's Characterization of the Lebesgue Measure. 3.9. Product Measures and Fubini's Theorem. 3.10. A Geometric Interpretation of the Integral. 3.11. Convolution Product. 3.12. The Radon-Nikodym Theorem. 3.13. Lebesgue Decomposition. 3.14. The Vitali-Hahn-Saks Theorem
4. Differentiation and Integration.

This text contains a basic introduction to the abstract measure theory and the Lebesgue integral. Most of the standard topics in the measure and integration theory are discussed. In addition, topics on the Hewitt-Yosida decomposition, the Nikodym and Vitali-Hahn-Saks theorems and material on finitely additive set functions not contained in standard texts are explored. There is an introductory section on functional analysis, including the three basic principles, which is used to discuss many of the classic Banach spaces of functions and their duals. There is also a chapter on Hilbert space and the Fourier transform.

https://www.worldscientific.com/worldscibooks/10.1142/2223