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Über dieses Buch

Integration theory and general topology form the core of this textbook for a first-year graduate course in real analysis. After the foundational material in the first chapter (construction of the reals, cardinal and ordinal numbers, Zorn's lemma and transfinite induction), measure, integral and topology are introduced and developed as recurrent themes of increasing depth. The treatment of integration theory is quite complete (including the convergence theorems, product measure, absolute continuity, the Radon-Nikodym theorem, and Lebesgue's theory of differentiation and primitive functions), while topology, predominantly metric, plays a supporting role. In the later chapters, integral and topology coalesce in topics such as function spaces, the Riesz representation theorem, existence theorems for an ordinary differential equation, and integral operators with continuous kernel function. In particular, the material on function spaces lays a firm foundation for the study of functional analysis.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Foundations

Abstract
The reader will already have a working familiarity with the concepts of set and function. Apart from a review of basic concepts and notations, the chapter is mostly about coping with infinity (or exploiting it-it depends−on one's point of view).
Sterling K. Berberian

Chapter 2. Lebesgue Measure

Abstract
One of the aims of the Lebesgue theory is to assign to each subset A of ℝ an element of [0, 1+∞], to be thought of as the ‘size’ of A, in such a way that the size of a bounded interval is its length, and the function
Sterling K. Berberian

Chapter 3. Topology

Abstract
The following informal remarks are intended as motivation for the subject of this chapter; the reader who already knows what a topological space is may prefer to plunge right into \(\S \) 3.1.
Sterling K. Berberian

Chapter 4. Lebesgue Integral

Abstract
Chapter 2 lays the foundation for the Lebesgue integral. What comes first is the concept of outer measure, a natural extension of interval length; the class of measurable sets is then defined in terms of the outer measure, guided by the need for additivity of measure as a set function. Outer measure provides the raw numbers; measurability provides the ‘smooth’ sets on which the numbers are well-behaved. Such well-behavior is codified in the concept of measure space.
Sterling K. Berberian

Chapter 5. Differentiation

Abstract
The focus of the chapter is on the “Fundamental theorem of calculus” for the Lebesgue theory, analogous to, but much harder than, the classical theorem of that name for the Riemann integral of a continuous function (the precise statements will be given shortly).
Sterling K. Berberian

Chapter 6. Function Spaces

Abstract
The emphasis of the present chapter is on metric spaces whose elements are functions (or equivalence classes of functions). The main examples arise in topological or measure-theoretic contexts; the first three sections prepare the way with the necessary topics in topology and metric spaces.
Sterling K. Berberian

Chapter 7. Product Measure

Abstract
If (X, S, μ) and (Y, T, v) are measure spaces, how can μ and v be combined to define a measure on a suitable σ-algebra of subsets of X x Y? The question is not academic: a satisfactory answer would open the door to constructing measures on ℝ2, ℝ3,..., starting with Lebesgue measure on ℝ. Lebesgue measure λ. on ℝ assigns to an interval [a, b] its length b - a; we expect the measure π on ℝ2 derived from λ to assign to a rectangle [a,b] x [c,d] its area (b-a)(d-c).
Sterling K. Berberian

Chapter 8. The Differential Equation y' = f(x, y)

Abstract
Our objective in this chapter is to solve the differential equation y' = f(x, y) for a suitable class of continuous real-valued functions f of two real variables x and y, where a “solution” is understood to be a real-valued (continuously differentiable) function φ : I → ℝ, defined on a suitable interval I, such that φ'(x) = f(x,φ(x)) for all x ϵ I.
Sterling K. Berberian

Chapter 9. Topics in Measure and Integration

Abstract
In Section 1, the decomposition of a finite signed measure as a difference of finite measures proved in Chapter 4 (4.8.8) is generalized to countably additive set functions admitting either +∞ or -∞ (but not both) as values.
Sterling K. Berberian

Backmatter

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