Measure and Integral: an Introduction to Real Analysis

Description

Content: Preliminaries Points and Sets in RnRn as a Metric SpaceOpen and Closed Sets in Rn: Special SetsCompact Sets The Heine-Borel TheoremFunctionsContinuous Functions and TransformationsThe Riemann IntegralExercises Function of Bounded Variation The Riemann-Stieltjes Integral Functions of Bounded VariationRectifiable CurvesThe Reiman-Stieltjes IntegralFurther Results About the Reimann-Stieltjes IntegralsExercises Lebesgue Measure and Outer Measure Lebesgue Outer Measures The Cantor Set. Lebesgue Measurable SetsTwo Properties of Lebesgue MeasureCharacterizations of MeasurabilityLipschitz Transformations of RnA Nonmeasurable Set. ExercisesLebesgue Measurable Functions Elementary Properties of Measurable Functions. Semicontinuous FunctionsProperties of Measurable Functions Egorov's Theorem and Lusin's TheoremConvergence in MeasureExercisesThe Lebesgue IntegralDefinition of the Integral of a Nonnegative FunctionProperties of the IntegralThe Integral of an Arbitrary Measurable f A Relation Between Riemann-Stieltjes and Lebesgue Integrals the LP Spaces, 0