Measure and Integration

A first course on measure theory and measure-theoretic integration, including the following topics: The Darboux formulation of the Riemann integral; Lebesgue outer measure on the real line; the sigma-algebra of Lebesgue measurable subsets of the real line; countable additivity; the Lebesgue integral; comparison of the Lebesgue and Riemann integrals; measurable functions; simple functions; measurable spaces; measures; integration of arbitrary non-negative measurable functions on a measure space; convergence of sequences of measurable functions and simple functions; the Monotone Convergence Theorem; integrability and integration of real-valued functions on a measure space; linearity of the integral; examples of integration over specific measure spaces; the Bounded Convergence Theorem; Lebesgue’s Dominated Convergence Theorem. Additional topics may include: Plane measure; product measure; Fubini’s theorem; signed measures and Hahn-Jordan decomposition; the Radon-Nikodym theorem.