📚 Real Analysis Notes

This repository contains comprehensive lecture notes for a Real Analysis course, covering topics such as set theory, measure theory, Lebesgue measurability, and Lebesgue integration in $\mathbb{R}^n$.

📋 Table of Contents

• Overview
• Document Structure
• Getting Started
  • Prerequisites
  • Compilation
• Dependencies
• Attribution
• License
• Contact

🔍 Overview

These notes were created based on a Real Analysis course at Ocean University of China. The content is designed for a 3-hour weekly course format and provides in-depth coverage of real analysis concepts. Please note that due to its specialized organization, these materials are not recommended for first-time learners of the subject. Also, to make the structure of real analysis much better, some part of materials that are not in the course would be put in.

📑 Document Structure

Covering Topics

The notes are organized into the following chapters:

1. Fundemantal Set Theory - Foundational concepts and theorems in set theory
2. Lebesgue Measure - Introduction to measure theory and Lebesgue measure
3. Measurable Functions - Properties and operations of measurable functions
4. Lebesgue Integration - Theory and applications of Lebesgue integration

Status

Section Number	Section Name	Status
1.1	Set Operations	✅
1.2	Set Equivalence	✅
1.3	Set Cardinality	✅
1.4	Metric Space	✅
1.5	Open Set, Closed Set and their Constructions	✅
1.6	Cantor Set	✅
2.1	Outer Measure	✅
2.2	Measurable Set and Measure	✅
2.3	Measurable Set Class	✅
2.4	Non-measurable Set	✅
2.5	Product Measure	✅
3.1	Measurable Function: Definition and Properties	✅
3.2	Measurable Function: Convergence	✅
3.3	Measurable Function: Continuous	✅
4.1	Lebesgue Integral: Non-negative Measurable Function	✅
4.2	Lebesgue Integral: All Measurable Function	✅
4.3	Lebesgue Integral and Riemann Integral	✅
4.4	Multiple Integral and Repeated Integral	✅
5	Tutorials	❇️

Notice that the whole notes is written in Chinese, the content might have some translation mistakes.

Status Symbol Meanings:

Status	Symbol
❎	Not started yet
❇️	Partially finished
✅	Already finished

🚀 Getting Started

Prerequisites

To work with these notes, you'll need:

• A LaTeX distribution (such as TeX Live or MiKTeX)
• Basic familiarity with LaTeX document preparation

Compilation

To compile this document:

# Using pdflatex
pdflatex Real Analysis Notes.tex

Alternatively, you can use LaTeX editors such as:

• TeXStudio
• Overleaf (online)
• VS Code with LaTeX Workshop extension

A much direct way to look through the notes is to just download the pdf version file.

📦 Dependencies

This document relies on the following resources:

• easybook.cls: Custom document class based on ElegantBook
• eb-elegantbook.sty: Core style file for the document
• Standard LaTeX Packages:
  • lmodern
  • codehigh
  • (and others as required)

👏 Attribution

This document uses a modified version of the ElegantBook LaTeX template, created by Ethan Deng and Liam Huang.

📬 Contact

For questions or suggestions regarding these notes:

• Email: [email protected]
• Issues: Please open an issue in this repository