Real Analysis: An Introduction to the Theory of Real Functions and Integration
Real Analysis: An Introduction to the Theory of Real Functions and Integration
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DOI：
10.1201/9781420036893
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发表时间：
2000-09
期刊：
影响因子：
--
通讯作者：
J. Dshalalow
中科院分区：
文献类型：
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作者：
J. Dshalalow
第一部分：一般拓扑学、集合论和代数预备、集和基本记号函数集在映射关系和良序原理下的集合运算、笛卡儿积基数基本代数结构分析度量空间的定义和符号度量空间的结构度量空间中的连续映射完备度量空间紧线性和赋范线性空间点集拓扑空间的元素拓扑空间的基和子基拓扑空间中序列的收敛和拓扑空间的可数性乘积拓扑学关于子空间和紧函数空间的拓扑学以及阿斯科利定理局部紧空间上的Stone-Weerstrass逼近定理滤子和网收敛分离函数测量与积分基础可测空间和可测函数集系统的生成元可测函数集函数集函数对测度的扩张勒贝格和勒贝格-斯蒂尔杰斯测度像测度扩展实值可测函数单函数积分在C-1上的元素(W，S)关于R积分关于积分生成的像测度的主要收敛定理。绝对的连续性。有限多个可测空间的正交性乘积测度与Fubini定理在欧氏空间中的应用微分变量变换第三部分抽象空间中积分分析的进一步主题符号与复测度绝对连续性奇性Lp空间收敛模式一致可积局部紧Hausdorff空间上的Radon测度有界变差绝对连续函数实线单调函数上的导数演算奇异函数
PART I. AN INTRODUCTION TO GENERAL TOPOLOGY SET-THEORETIC AND ALGEBRAIC PRELIMINARIES Sets and Basic Notation Functions Set Operations under Maps Relations and Well-Ordering Principle Cartesian Product Cardinality Basic Algebraic Structures ANALYSIS OF METRIC SPACES Definitions and Notations The Structure of Metric Spaces5 Convergence in Metric Spaces Continuous Mappings in Metric Spaces Complete Metric Spaces Compactness Linear and Normed Linear Spaces ELEMENTS OF POINT SET TOPOLOGY Topological Spaces Bases and Subbases for Topological Spaces Convergence of Sequences in Topological Spaces and Countability Continuity in Topological Spaces Product Topology Notes on Subspaces and Compactnes Function Spaces and Ascoli's Theorem Stone-Weierstrass Approximation Theorem Filter and Net Convergence Separation Functions on Locally Compact Spaces PART II. BASICS OF MEASURE AND INTEGRATION MEASURABLE SPACES AND MEASURABLE FUNCTIONS Systems of Sets System's Generators Measurable Functions MEASURES Set Functions Extension of Set Functions to a Measure Lebesgue and Lebesgue-Stieltjes Measures Image Measures Extended Real-Valued Measurable Functions Simple Functions ELEMENTS OF INTEGRATION Integration on C -1(W,S) Main Convergence Theorems Lebesgue and Riemann Integrals on R Integration with Respect to Image Measures Measures Generated by Integrals. Absolute Continuity. Orthogonality Product Measures of Finitely Many Measurable Spaces and Fubini's Theorem Applications of Fubini's Theorem CALCULUS IN EUCLIDEAN SPACES Differentiation Change of Variables PART III. FURTHER TOPICS IN INTEGRATION ANALYSIS IN ABSTRACT SPACES Signed and Complex Measures Absolute Continuity Singularity Lp Spaces Modes of Convergence Uniform Integrability Radon Measures on Locally Compact Hausdorff Spaces Measure Derivatives CALCULUS ON THE REAL LINE Monotone Functions Functions of Bounded Variation Absolute Continuous Functions Singular Functions BIBLIOGRAPHY