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 1 Measure on a σ-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 L^p Spaces 17 Relation among the L^p Spaces 18 Bounded Linear Functionals on the L^p Spaces 22 Lebesgue-Stieltjes Measure Spaces 23 Product Measure Spaces 24 Lebesgue Measure Space on the Euclidean Space 25 Differentiation on the Euclidean Space