Measure Theory

I used this book for my graduate measure theory class. We covered the first seven chapters. The book is written in a clear fashion and is easy to follow. It is concise and at the same time is almost self contained.The first chapter gives an introduction to measure theory. It deals with sigma algebras, measures, outer measures, completeness and regularity. The lebesgue measure is also introduced in this chapter.The second chapter starts of measurable functions. It then proceeds to almost sure properties followed by integration. These are followed by the theorem: Monotone convergence theorem, Beppo Levi theorem, Fatou's Lemma and Dominated Convergence Theorem. The chapter also discusses briefly on Riemann integrals.The third chapter is on different modes of convergence. It proves the Egoroff theorem. This is followed by the definition and properties of Banach spaces.The fourth chapter discusses signed measures. The Hahn decomposition theorem and Jordan decomposition theorems are proved. It is followed by absolutely continuous measures which leads to Radon-Nikodym theorem.The fifth chapter deals with Product measures. The most important theorem in this chapter is the Fubini's theorem which is proved in the second section.The sixth chapter is on differentiation of measures. Proves Fundamental theorem of calculus.The seventh chapter is on Hausdorff spaces and Riesz representation theorem followed by properties of regular measures (Lusin's theorem)Chapter 8 is on Polish Spaces and Analytic SetsChapter 9 is on Haar Measures(We did not cover the last two chapters in this course)