Real AnalysisWorld Scientific, 2025 M01 10 - 212 páginas Can the limitations of the Riemann integral be overcome? What is its relationship with modern analysis?The theory of Lebesgue integration is a crucial component in the development of modern analysis. This book is an in-depth real analysis textbook, which introduces the basic theory of modern analysis and the basic skills of analysis. Based on the knowledge of real analysis, the theory of interpolation of operators and the Fourier transform theory are further introduced systematically. The main contents include: abstract measures and integrals, measure and topology, Lebesgue integration on Rn, the interpolation of operators on Lp(Rn), Hardy-Littlewood maximal function, convolution and the Fourier transform. They play an important role in harmonic analysis, partial differential equations, probability and numerical analysis. This book is moderately difficult and detailed, focusing on the combination of abstract and concrete, and training readers to skillfully use modern analysis.This textbook is an excellent reference book for readers studying the fields of Harmonic analysis and partial differential equations. It is intended for advanced undergraduate and graduate students in university mathematics, as well as mathematicians and physicists in general. |
Contenido
| 1 | |
2 Measure and Topology | 69 |
3 Lebesgue Integration on Rn | 119 |
4 The Interpolation of Operator on LpRn | 137 |
5 HardyLittlewood Maximal Function | 155 |
6 Convolution | 167 |
7 The Fourier Transform | 181 |
| 195 | |
| 197 | |
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A-measurable Borel algebra called card(A Cc(X closed set compact set conclusion continuous function convergence Corollary cube Define Definition denoted exists an open finite measure following theorem Fourier transform Fubini's theorem function f functions of real Haar measure Hausdorff space Hölder's inequality holds implies K₁ L¹(R L¹(Rn L²(R L²(Rn LCHS Lebesgue integral left Haar measure Lemma Let f Let G Let µ locally compact group LP(Rn LP(X mapping measurable function measurable rectangles measurable set measure on G measure space nonempty o-algebra o-finite Obviously open set outer measure pairwise disjoint premeasure Proof Radon measure real variable real-valued respect Rn Rn satisfying sequence signed measure simple functions subset supp f Suppose theory of functions topological group topological space usual Lebesgue measure v₁ weak type μο Σμ