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## Über dieses Buch

This compact textbook is a collection of the author’s lecture notes for a two-semester graduate-level real analysis course. While the material covered is standard, the author’s approach is unique in that it combines elements from both Royden’s and Folland’s classic texts to provide a more concise and intuitive presentation. Illustrations, examples, and exercises are included that present Lebesgue integrals, measure theory, and topological spaces in an original and more accessible way, making difficult concepts easier for students to understand. This text can be used as a supplementary resource or for individual study.

## Inhaltsverzeichnis

### Chapter 1. Measures

Abstract
To understand why we need a theory of measures and a new way to define integrals (Lebesgue integrals), we need to look what is wrong with Riemann’s theory of integrals.
Xiaochang Wang

### Chapter 2. Integration

Abstract
Recall that in order to define the Lebesgue integral for a function f, the set
Xiaochang Wang

### Chapter 3. Signed Measures and Differentiation

Abstract
Let $$(X,{\mathcal M})$$ be a measurable space. A signed measure of $$(X,{\mathcal M})$$ is a countably additive set function $$\nu :{\mathcal M}\to [-\infty ,\infty )$$ or (−, ] such that ν(∅) = 0.
Xiaochang Wang

### Chapter 4. Topology: A Generalization of Open Sets

Abstract
A collection $${\mathcal T}$$ of subsets of X is called a topology on X if
Xiaochang Wang

### Chapter 5. Elements of Functional Analysis

Abstract
Let X be a vector space over $${\mathbb R}$$. A norm ∥⋅∥ on X is a function X → [0, ) such that
Xiaochang Wang

### Chapter 6. Lp Spaces

Abstract
Let $$(X,{\mathcal M},\mu )$$ be a measure space and p > 0.
Xiaochang Wang

### Backmatter

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