Measure Theory

1. Measures. - Algebras and sigma-algebras. - Measures. - Outer measures. - Lebesgue measure. - Completeness and regularity. - Dynkin classes. - 2. Functions and Integrals. - Measurable functions. - Properties that hold almost everywhere. - The integral. - Limit theorems. - The Riemann integral. - Measurable functions again, complex-valued functions, and image measures. - 3. Convergence. - Modes of Convergence. - Normed spaces. - Definition of L^p and L^p. - Properties of L^p and L-p. - Dual spaces. - 4. Signed and Complex Measures. - Signed and complex measures. - Absolute continuity. - Singularity. - Functions of bounded variation. - The duals of the L^p spaces. - 5. Product Measures. - Constructions. - Fubini s theorem. - Applications. - 6. Differentiation. - Change of variable in R^d. - Differentiation of measures. - Differentiation of functions. - 7. Measures on Locally Compact Spaces. - Locally compact spaces. - The Riesz representation theorem. - Signed and complex measures; duality. - Additional properties of regular measures. - The µ^*-measurable sets and the dual of L^1. - Products of locally compact spaces. - 8. Polish Spaces and Analytic Sets. - Polish spaces. - Analytic sets. - The separation theorem and its consequences. - The measurability of analytic sets. - Cross sections. - Standard, analytic, Lusin, and Souslin spaces. - 9. Haar Measure. - Topological groups. - The existence and uniqueness of Haar measure. - The algebras L^1 (G) and M (G). - Appendices. - A. Notation and set theory. - B. Algebra. - C. Calculus and topology in R^d. - D. Topological spaces and metric spaces. - E. The Bochner integral. - F Liftings. - G The Banach-Tarski paradox. - H The Henstock-Kurzweil and McShane integralsBibliography. - Index of notation. - Index.