Real Analysis Lecture 0 Introduction

Posted on Edited on In ,
Lecture Notes for 0.Introduction to Real Analysis.

This course aims to prepare for functional analysis and is composed by 2 parts: 1. Integration (the sum of (weighted) function values on \([a,b]\)) (measure, Lebesgue integration, …) 2. Differentiation (from the perspective of integration) (maximal function, covering lemma)

Integration

“a integration theory that is comparable to absolute summation”
that is, it inherits the property of absolutely convergent series.

Defects of Riemann Integration

\[ \mathrm{R}(f,\Delta,\{\xi_i\}) = \sum_{i=1}^{n}(x_i - x_{i-1}) \cdot f(\xi_i) \] and \[ \int_a^b f(x)\,dx := \lim_{|x_i - x_{i-1}| \rightarrow 0} \mathrm{R}(f,\Delta,\{\xi_i\}) \] does not describe the theory we want because it cares the sum order: the function can’t jump/oscillate too frequently.

Lebesgue’s Method

Recall: if \(\sum a_n\) converges absolutely, then \(\sum a_n = \sum a_n^+ + \sum a_n^-\)
comparably, we want that: \[ \int_a^b f(x)\,dx = \int_a^b f(x)^+\,dx + \int_a^b f(x)^-\,dx \]

Or more generally, partition on \(Y(\text{range } f)\): \(\Delta: y_0 < y_1 < \cdots < y_i < \cdots\)
Define: \[ \mathrm{L}(f,\Delta,\{y_i\}) = \sum_{i=1}^n y_i \cdot \mu\left(f^{-1}([y_i,y_{i+1}])\right) \] but \(f^{-1}([y_i,y_{i+1}])\) is non-trivial and is reasonable enough to be given a notion of length.

Remark: the action/notion determines the set. {: .prompt-info}

Differentiation

“from the perspective to integration”
Let \(F:[a,b]\rightarrow\mathbb{R}\), \(F\uparrow\).
For \([\alpha,\beta] \subset [a,b]\), we may assign a F-length: \(F(\beta) - F(\alpha)\), and define F-integral (a functional in fact): \[ \varphi \stackrel{\mathrm{L}}{\longmapsto} \int_a^b \varphi(x)\,dF \] We wonder whether this functional can be represented by a function \(f\) in the sense that: \[ \mathrm{L}(\varphi) = \int_a^b \varphi(x)f(x)\,dx, \] where \(dx\) is the original measure.
Then \(f\) is a “good” derivative of \(F\).

Question: if \(f\) exists almost everywhere, could it recover \(F\)? (As Riemann’s integral does)
Answer: Not necessary. {: .prompt-warning}

Cardinality

Thm: (Cantor-Bernstein) \(A,B\) are sets, if \(|A| \leq |B|\) and \(|B| \leq |A|\) then \(|A| = |B|\)