Type of Credit: Partially Required
Credit(s)
Number of Students
Measurable Spaces
- Algebra of Sets
- sigma-slgebra of Sets
- Borel Fields
- Monotone Classes
Measure Spaces
- Basic Properties of Measure
- A class of Measures
- Completion of Measure Space
Construction of Meausre
- Outer Measure
- Theorem of Caratheodory
- A Method of Constructing Outer Measure
- Meausre on Algebra
- Extension Theorem
Lebesgue Measure on R^n
- Some Preliminaries
- The n-dimensional Lebesgue Outer Measure
- Properties of Lebesgue Measure Space
- Invariant Properties of Lebesgue Measure
- An Uncountable Lebesgue Null Set - Cantor Set
- A Nonmeasurable Set
Borel Measures
- Some Preliminaries
- Properties of Monotone Function
- Lebesgue-Stieltjes Measures on R
- Borel Measure on Topological Space
Measurable Functions
- Some Preliminaries
- Measurable Function
- Operations on Measurable Functions
- Further Measurability on Funcions
- Simple Functions
- The Role of Null Sets
Integration
- Integral of Nonnegative Simple Function
- Integral of Nonnegative Measurable Function
- Lebesgue Monotone Convergence Theorem
- Fatou's Lemma
- Integral of General Measurable Function
- Lebesgue Dominated Convergence Theorem
- Integral over Measurable Subsets
- Point-mass Distributions
- Lebesgue Integral
- Relation between Riemann and Lebesgue Integral
- Lebesgue-Stieltjes Integral
- Reduction to Integrals over R
能力項目說明
本課程之目標在建立學生分析的基礎作為未來相關領域之發展
| 教學週次Course Week | 彈性補充教學週次Flexible Supplemental Instruction Week | 彈性補充教學類別Flexible Supplemental Instruction Type |
|---|---|---|
| 次Week | 課程主題Course Theme | 課程內容與指定閱讀Content and Reading Assignment | 教學活動與作業Activity and Homework | 學習投入時數Estimated time devoted to coursework per week | |
|---|---|---|---|---|---|
| 課堂講授Lecture Hours | 課程前後Preparation Time | ||||
|
1 |
Measurable Spaces |
Algebra of Sets, sigma-algebra of Sets |
Exercisrs in chapter1 and discussion |
3.0 |
4.5 |
|
2 |
Measurable Spaces |
Algebra of Sets, sigma-algebra of Sets |
Exercisrs in chapter1 and discussion |
3.0 |
4.5 |
|
3 |
Measurable Spaces |
Borel Fields, Monotone Classes |
Exercisrs in chapter1 and discussion |
3.0 |
4.5 |
|
4 |
Measure Spaces |
Basic Properties |
Exercisrs in chapter2 and discussion |
3.0 |
4.5 |
|
5 |
Measure Spaces |
Completion of Measure Spaces |
Exercisrs in chapter2 and discussion |
3.0 |
4.5 |
|
6 |
Construction of Meausre |
Outer Measures, Theorem of Caratheodory |
Exercisrs in chapter3 and discussion |
3.0 |
4.5 |
|
7 |
Construction of Meausre |
Meausre on Algebra, Extension Theorem |
Exercisrs in chapter3 and discussion |
3.0 |
4.5 |
|
8 |
Lebesgue Measure on R^n |
The n-dimensional Lebesgue Outer Measure and its Properties |
Exercisrs in chapter4 and discussion |
3.0 |
4.5 |
|
9 |
Midterm |
Midterm |
Midterm |
3.0 |
4.5 |
|
10 |
Lebesgue Measure on R^n |
An Uncountable Lebesgue Null Set - Cantor Set, Nonmeasurable Set |
Exercisrs in chapter4 and discussion
|
3.0 |
4.5 |
|
11 |
Borel Measures |
Lebesgue-Stieltjes Measures on R |
Exercisrs in chapter5 and discussion |
3.0 |
4.5 |
|
12 |
Borel Measures |
Borel Measure on Topological Space |
Exercisrs in chapter5 and discussion |
3.0 |
4.5 |
|
13 |
Measurable Functions |
Operations on Measurable Functions |
Exercisrs in chapter6 and discussion |
3.0 |
4.5 |
|
14 |
Measurable Functions |
Simple Functions |
Exercisrs in chapter6 and discussion |
3.0 |
4.5 |
|
15 |
Integration |
Integral of Nonnegative Simple, Nonnegative Measurable Functions |
Exercisrs in chapter7 and discussion |
3.0 |
4.5 |
|
16 |
Integration |
Lebesgue Monotone Convergence Theorem, Fatou's Lemma, Integral of General Measurable Functions, Lebesgue Dominated Convergence Theorem |
Exercisrs in chapter7 and discussion |
3.0 |
4.5 |
|
17 |
Integration |
Integral over Measurable Subsets, Relation between Riemann and Lebesgue Integral, Lebesgue-Stieltjes Integral |
Exercisrs in chapter7 and discussion |
3.0 |
4.5 |
|
18 |
Final |
Final |
|||
成績評量:
期中考成績: 40%
期末考成績：60%
學生有任何建議可於開學上課時提出。
Note on measure theory by M. Papadimitrakis
References:
1. Real Analysis, 3rd edition, by H.L. Royden.
2. Principles of Real Analysis, 2nd edition, by C. D. Aliprantis and O. Burkinshaw.
3. Measure and Integral by R. L. Wheeden and A. Zygmund.
4. The Elements of Integration and Lebesgue Measure by R. G. Bartle
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