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課程大綱及進度表

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課程大綱及進度表

開課系所

數學系

開課學年

101

開課學期

1

課程名稱(中文)

實變數函數論

課程名稱(英文)

Real Analysis

課程碼

C136200

分班碼

先修科目或先備能力

學分數

3

開課教師

吳順益

e-mail

soonyi@mail.ncku.edu.tw

電話

(06)2757575 轉 65133

Office Hours

By Appointment

課程概述

主要討論 Lebesque Measure、Lebesque Measurable Function 及 Lebesque Integration,主要是教導實變函數的基 礎概念。

教學目標

主要是讓選這門課程的學生對 Lebesque Measure,與 Lebesque Integration 有 一些基礎概念。

授課課程大綱明細

Chapter 1. Lebesgue measure

1.1 Introduction

1.2 Lebesgue other measure

1.3 The ‐ Algebra of Lebesque Measure Sets

1.4 Outer and Inner Approximation of Lebesque

Measurable Sets

1.5 Countable Additivity, Continuity, and

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the

Borel‐Contelli Lemma

Chapter 2. Lebesque Measurable Functions

2.1 Sums, Products, and Compositions 2.2 Sequential Pointwise Limits and Simple Approximation 2.3 Littlewood’s Three Principles, Egoroff’s

Theorem, and Lusin’s Theorem Chapter3. Lebesque Integration 3.1 The Riemann Integral

3.2 The Lebesuqe Integral of a Bounded Measurable Nonnegative Function over a set of

Finite Measure

3.3 The Lebesque Integral of a Measurable Nonnegative Function

3.4 The General Lebesque Integral 3.5 Countable Additivity and Continuity of Integration

3.6 Uniform Integrability: The Vitali Convergence

Theorem

Chapter4. Lebesque Integration:

Further Topics

4.1 Uniform Integrability and Tightness: A General Vitali Convergence Theorem 4.2 Convergence in Measure

4.3 Characterizations of Riemann and Lebesque

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Integrability

參考書目 Real Analysis

(Fourth Edition) by Halsey Royden and Patrick

Fitzpatrick

課程要求

評量方式

期中考:40%

期末考:40%

平時成績:20%

課程網址

助教資訊

備註

參考文獻

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