A Course in Analysis Vol. III: Measure and Integration Theory, Complex-Valued Functions of a Complex Variable

In this third volume of "A Course in Analysis", two topics indispensible for every mathematician are treated: Measure and Integration Theory; and Complex Function Theory. In the first part measurable spaces and measure spaces are introduced and Caratheodory's extension theorem is proved. This is followed by the construction of the integral with respect to a measure, in particular with respect to the Lebesgue measure in the Euclidean space. The Radon–Nikodym theorem and the transformation theorem are discussed and much care is taken to handle convergence theorems with applications, as well as Lp-spaces. Integration on product spaces and Fubini's theorem is a further topic as is the discussion of the relation between the Lebesgue integral and the Riemann integral. In addition to these standard topics we deal with the Hausdorff measure, convolutions of functions and measures including the Friedrichs mollifier, absolutely continuous functions and functions of bounded variation. The fundamental theorem of calculus is revisited, and we also look at Sard's theorem or the Riesz–Kolmogorov theorem on pre-compact sets in Lp-spaces. The text can serve as a companion to lectures, but it can also be used for self-studying. This volume includes more than 275 problems solved completely in detail which should help the student further. Contents: Measure and Integration Theory:First Look at σ-Fields and MeasuresExtending Pre-Measures. Carathéodory's TheoremThe Lebesgue-Borel Measure and Hausdorff MeasuresMeasurable MappingsIntegration with Respect to a Measure — The Lebesgue IntegralThe Radon-Nikodym Theorem and the Transformation TheoremAlmost Everywhere Statements, Convergence TheoremsApplications of the Convergence Theorems and MoreIntegration on Product Spaces and ApplicationsConvolutions of Functions and MeasuresDifferentiation RevisitedSelected TopicsComplex-Valued Functions of a Complex Variable:The Complex Numbers as a Complete FieldA Short Digression: Complex-Valued MappingsComplex Numbers and GeometryComplex-Valued Functions of a Complex VariableComplex DifferentiationSome Important FunctionsSome More TopologyLine Integrals of Complex-Valued FunctionsThe Cauchy Integral Theorem and Integral FormulaPower Series, Holomorphy and Differential EquationsFurther Properties of Holomorphic FunctionsMeromorphic FunctionsThe Residue TheoremThe Γ-Function, The ζ-Function and Dirichlet SeriesElliptic Integrals and Elliptic FunctionsThe Riemann Mapping TheoremPower Series in Several VariablesAppendices:More on Point Set TopologyMeasure Theory, Topology and Set TheoryMore on Möbius TransformationsBernoulli Numbers Readership: Undergraduate students in mathematics.