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.

From the book reviews:

"This textbook provides a comprehensive and consistent introduction to measure and integration theory. ... The book can be recommended to anyone having basic knowledge of calculus and point-set topology. It is very self-contained, and can thus serve as an excellent reference book as well." (Ville Suomala, Mathematical Reviews, July, 2014)

"In this second edition, Cohn has updated his excellent introduction to measure theory ... and has made this great textbook even better. Those readers unfamiliar with Cohn's style will discover that his writing is lucid. ... this is a wonderful text to learn measure theory from and I strongly recommend it." (Tushar Das, MAA Reviews, June, 2014)