Problems and proofs in real analysis: theory of measure and integration

Description

This volume consists of the proofs of 391 problems in Real Analysis: Theory of Measure and Integration (3rd Edition). Most of the problems in Real Analysis are not mere applications of theorems proved in the book but rather extensions of the proven theorems or related theorems. Proving these problems tests the depth of understanding of the theorems in the main text. This volume will be especially helpful to those who read Real Analysis in self-study and have no easy access to an instructor or an advisor Contents Preface §1 Measure on a a-algebra of Sets §2 Outer Measures §3 Lebesgue Measure on R §4 Measurable Functions §5 Completion of Measure Space §6 Convergence a.e. and Convergence in Measure §7 Integration of Bounded Functions on Sets of Finite Measure §8 Integration of Nonnegative Functions §9 Integration of Measurable Functions § 10 Signed Measures § 11 Absolute Continuity of a Measure § 12 Monotone Functions and Functions of Bounded Variation §13 Absolutely Continuous Functions § 16 The LP Spaces § 17 Relation among the LP Spaces § 18 Bounded Linear Functionals on the LP Spaces §22 Lebesgue-Stieltjes Measure Spaces §23 Product Measure Spaces §24 Lebesgue Measure Space on the Euclidean Space §25 Differentiation on the Euclidean Space