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所在平台: Udemy |
课程主页: https://www.udemy.com/course/lebesgue-measure-part-1/
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**课程名称：**实分析第五部分（勒贝格测度） **课程概述：** 本课程是“勒贝格测度”系列的第二部分，重点介绍“可测集和勒贝格测度”。勒贝格测度是由法国数学家亨利·勒贝格命名的一种数学分支，它是为 n 维欧几里得空间子集分配测度的标准方法。可测集被定义为在该系统中可以实现测度扩展的集合。 **课程内容：** 本课程将涵盖以下主题： * **集合代数 (The Algebra of Sets)** * **波莱尔集 (Borel Sets)** * **外测度 (Outer Measure)** * 集合的外测度 * 区间的**外测度** * **可数次可加性 (Countable Subadditivity)**：一个在许多定理和命题中被广泛应用的重要性质。 * **可测集 (Measurable Sets)** * 可测集的族 (Family of Measurable sets) * 不可测集的存在性 (Existence of a Non Measurable set) * **勒贝格测度 (Lebesgue Measure)** * 勒贝格测度的定义及其在**两两不交可测集**中的重要性。 * 可测集的**无限递减序列**。 * 一个证明勒贝格测度在**模 1 的平移下不变性**的引理。 * **可测函数 (Measurable Functions)** * 具有可测域的**扩展实值函数**及其等价性质。 * 两个具有相同定义域的实值可测函数，我们将学习它们的**和、差、积**仍然是可测函数的**结果**。 * **利特尔伍德三个原理 (LITTLEWOOD'S THREE PRINCIPLES)**：一个非常重要的原理。 课程将涵盖以上所有内容相关的**重要定理、命题、引理以及例题讲解**。 **教学大纲：** 无
LEBESGUE MEASURE Part 1'Measurable Sets and Lebesgue Measure'In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.And, A measurable set was defined to be a set in the system to which the extension can be realized; this extension is said to be the measure.Contents of the Course:In this Course you will learn about ' The Algebra of Sets' , 'Borel Sets' including ' 'Outer Measure'_ Outer measure of a set _ Outer measure of an intervalA very important property ' Countable Subadditivity' is almost used in many theorems and Propositions. Then you will also learn about Measurable Sets, Family of Measurable sets and also the Existence of a Non Measurable set. Next, the Definition of Lebesgue Measure and its importance in pairwise disjoint measurable sets, Infinite decreasing sequence of Measurable Sets. The Lemma that shows that Lebesgue measure is invariant under Translation modulo 1.And, Finally at last ; you will learn about 'Measurable Functions' and its importance in extended real valued function having measurable Domain with Equivalent Properties. Also if two measurable real valued functions are having same domain with constant then you will learn the results that the sum, difference, product of these functions are also measurable, including a very Important Principles called 'LITTLEWOOD'S THREE PRINCIPLES'.All the important theorems, Propositions, Lemma and Solved Examples are covered based on all above mentioned contents.Thanks.
