Measure Theory and IntegrationHorwood Publishing, 15 lug 2003 - 240 pagine This text approaches integration via measure theory as opposed to measure theory via integration, an approach which makes it easier to grasp the subject. Apart from its central importance to pure mathematics, the material is also relevant to applied mathematics and probability, with proof of the mathematics set out clearly and in considerable detail. Numerous worked examples necessary for teaching and learning at undergraduate level constitute a strong feature of the book, and after studying statements of results of the theorems, students should be able to attempt the 300 problem exercises which test comprehension and for which detailed solutions are provided.
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Sommario
Preface | 9 |
Preliminaries | 15 |
Measure on the Real Line | 27 |
Integration of Functions of a Real Variable | 54 |
Differentiation | 77 |
Abstract Measure Spaces | 93 |
Inequalities and the LP Spaces | 109 |
Convergence | 121 |
Signed Measures and their Derivatives | 133 |
LebesgueStieltjes Integration | 153 |
Measure and Integration in a Product Space | 176 |
Hints and Answers to Exercises | 197 |
References | 236 |
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A₁ absolutely continuous B₁ Borel measurable Borel sets Cantor-like set Chapter Clearly contains continuous function convergence convex Corollary Definition disjoint sets E₁ E₂ equality ess sup Example 12 exists f₁ f₂ Fatou's Lemma finite interval finite measure finite-valued fn(x function ƒ ƒ and g ƒ ƒ dx ƒ is integrable ƒ is measurable gives the result Hahn decomposition Hausdorff measure hence Hölder's inequality indefinite integral inequality integrable function last exercise Lebesgue measure left-continuous Let f Let ƒ lim inf lim sup measurable function measurable set measurable simple functions measure space measure zero non-measurable set non-negative measurable functions notation o-algebra o-finite o-ring obtain open intervals open set outer measure Proof real numbers result follows sequence of measurable Show signed measure Similarly Solution step function subsets suppose Theorem 11 Theorem 9 v₁ v₂ write