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所在平台: Udemy |
课程主页: https://www.udemy.com/course/the-lebesgue-integral/
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## Coursera 课程总结：实分析（六）——勒贝格积分 本课程深入探讨了数学分析中的核心概念——勒贝格积分，该积分由法国数学家亨利·勒贝格提出，是对黎曼积分的重大扩展。 **核心内容概览：** * **勒贝格积分的产生与优势：** 勒贝格积分能够处理比黎曼积分更广泛的函数类和定义域，尤其能积分黎曼积分甚至黎曼-斯蒂尔切斯积分无法处理的函数。 * **积分思想的转变：** 课程解释了勒贝格积分的核心思想：传统的积分方法是将积分区域划分成垂直的矩形（如黎曼积分），而勒贝格积分则是将积分区域划分成水平的“薄片”，这些薄片不必是规则的矩形。 * **广阔的应用前景：** 勒贝格积分在概率论、实分析及傅里叶级数、傅里叶变换等多个数学领域扮演着至关重要的角色。 * **极限运算的便利性：** 勒贝格积分在处理积分号下的极限运算（通过单调收敛定理和支配收敛定理）方面表现出优越性。 **课程涵盖的具体知识点：** * **阶梯函数介绍** * **特征函数和简单函数的定义** * **有界函数在有限测度集上的勒贝格积分** * **函数可测性的必要与充分条件** * **勒贝格积分的定义** * **勒贝格可积但黎曼不可积的函数** * **勒贝格积分的性质** * **有界收敛定理** * **非负函数的积分** * **法图引理** * **单调收敛定理及其推论** * **定义在可测集上的非负可积函数** * **课程包含了所有相关的命题、定理和引理。** **课程时长：** 3小时7分钟。 “Real Analysis Part 6 (THE LEBESGUE INTEGRAL)” 是一门为希望深入理解现代分析工具的学生设计的课程，它将为数学研究的各个领域打下坚实的基础。
The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined. The Lebesgue integrals are the integration of functions over measurable sets, which could integrate many functions that cannot be integrated as Riemann integrals or even Riemann-Stieltjes integrals. The concept behind the Lebesgue integrals is that generally, while integrating a given function, the total area under the curve is divided into several vertical rectangles, but while determining the Lebesgue integral of the function, the area under the curve is divided into horizontal slabs, that need not be rectangles. The Lebesgue Integral plays an important role in Probability theory, Real Analysis, and many other fields in Mathematics. In the study of Fourier series, Fourier transforms, and other topics. The Lebesgue integral is better able to describe how and when it is possible to take limits under the integral sign (via the Monotone Convergence Theorem and Dominated Convergence Theorem).This Impressive Course of 3 hr 7 min includes the Contents_Introduction of Step FunctionDefinition of Characteristic Function & Simple FunctionsThe Lebesgue Integral of a Bounded Function over a set of Finite Measure.Necessary and Sufficient Condition for a function to be Measurable.Definition of a LEBESGUE INTEGRALFunction that is Lebesgue Integral but not Riemann Integral.Properties of Lebesgue Integrals.BOUNDED CONVERGENCE THEOREMThe Integral of a Non-Negative FunctionFATOU'S LEMMAMONOTONE CONVERGENCE THEOREM Corollary of Monotone Convergence TheoremDefinition of a Non Negative Function Integrable over the Measurable SetIncluding all Propositions, Theorems and Lemma's.Thank You.
