Measure and Integral: An Introduction to Real AnalysisThis volume develops the classical theory of the Lebesgue integral and some of its applications. The integral is initially presented in the context of n-dimensional Euclidean space, following a thorough study of the concepts of outer measure and measure. A more general treatment of the integral, based on an axiomatic approach, is later given. Closely related topics in real variables, such as functions of bounded variation, the Riemann-Stieltjes integral, Fubini's theorem, L(p)) classes, and various results about differentiation are examined in detail. Several applications of the theory to a specific branch of analysis--harmonic analysis--are also provided. Among these applications are basic facts about convolution operators and Fourier series, including results for the conjugate function and the Hardy-Littlewood maximal function. Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis for student interested in mathematics, statistics, or probability. Requiring only a basic familiarity with advanced calculus, this volume is an excellent textbook for advanced undergraduate or first-year graduate student in these areas. |
Contents
Introduction | 1 |
Chapter | 4 |
Functions of Bounded Variation the Riemann | 15 |
Rectifiable Curves | 21 |
Further Results About RiemannStieltjes Integrals | 28 |
Lebesgue Measurable Functions | 50 |
3 | 56 |
The Lebesgue Integral | 64 |
Chapter 6 | 93 |
The Indefinite Integral | 99 |
The Vitali Covering Lemma | 109 |
Absolutely Continuous and Singular Functions | 115 |
Convex Functions | 121 |
Approximations of the Identity Maximal Functions | 145 |
Abstract Integration | 161 |
Outer Measure Measure | 193 |
The Integral of an Arbitrary Measurable | 71 |
Riemann and Lebesgue Integrals | 80 |
Repeated Integration | 87 |
A Few Facts From Harmonic Analysis | 211 |
Notation | 265 |
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a₁ absolutely continuous additive set function Borel measure Borel set bounded variation Chapter choose closed set compact completes the proof constant continuous on a,b convergence theorem corollary countable cubes denote disjoint E₁ E₂ example Exercise exists f(xo f₁ f₂ fact formula Fourier coefficients Fubini's theorem function defined function ƒ ƒ and g ƒ is continuous ƒ is measurable given Hence Hölder's inequality indefinite integral interval a,b Lebesgue integral Lebesgue measure lemma Let f Let ƒ liminf limsup linear measurable functions measurable sets measurable subset measure zero nonnegative and measurable obtain open intervals open set outer measure prove result Riemann integral Riemann-Stieltjes integral satisfies SEƒ sequence space subinterval Suppose Theorem If f Theorem Let variation on a,b x₁ π π
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Page ix - ... properties (i) ||x|| > 0 and ||x|| = 0 if and only if x = 0. (ii) ||ax|| = a| ||x||, where |a| is the absolute value of the complex scalar a.