Integration Advanced

Advanced (Graduate level)

The Lebesgue Integral and $L^p$ Spaces

Overview

Lebesgue Measure $\sigma$-algebra Measurable sets Null sets Lebesgue Integral Simple-function approximation $\int f \, d\mu$ Nonnegative → general Convergence Theorems Monotone convergence Fatou's lemma Dominated convergence $L^p$ Spaces $\|f\|_p = \left(\int |f|^p\right)^{1/p}$ Completeness Hölder, Minkowski Lebesgue integral: "partition the range" → handles a broader class of functions

At the advanced level, we study the Lebesgue integral, the foundation of modern analysis. We treat the rigorous definition of integration based on measure theory, the powerful convergence theorems, and the $L^p$ spaces that are central to functional analysis.

Learning Goals

  • Understand Lebesgue measure and measurable sets
  • Understand the construction of the Lebesgue integral
  • Master the convergence theorems (monotone convergence, dominated convergence)
  • Understand the structure of $L^p$ spaces

Table of Contents

  1. Chapter 1 $\sigma$-Algebras and Measures

    $\sigma$-algebras, definition of a measure, completion

  2. Chapter 2 Lebesgue Measure

    Measure on $\mathbb{R}^n$, outer measure, Carathéodory

  3. Chapter 3 Measurable Functions

    Measurable functions, simple functions, almost everywhere

  4. Chapter 4 The Lebesgue Integral

    Nonnegative functions, general functions, relation to Riemann

  5. Chapter 5 Convergence Theorems

    Monotone convergence, Fatou, dominated convergence, interchange of integral and limit

  6. Chapter 6 $L^p$ Spaces

    $L^p$ norm, completeness, density, duality

Prerequisites

  • Content of Integration Intermediate
  • Basics of topological spaces
  • Basics of set theory (cardinality, the axiom of choice)

Key Theorems

Monotone Convergence Theorem

If $0 \leq f_1 \leq f_2 \leq \cdots$ is a sequence of measurable functions with $f_n \to f$ (pointwise convergence), then:

$$\lim_{n \to \infty} \displaystyle\int f_n \, d\mu = \displaystyle\int f \, d\mu$$

Dominated Convergence Theorem (Lebesgue)

If $f_n \to f$ (pointwise convergence) and $|f_n| \leq g$ (with $g$ integrable), then:

$$\lim_{n \to \infty} \displaystyle\int f_n \, d\mu = \displaystyle\int f \, d\mu$$

Completeness of $L^p$ Spaces

$L^p(\mu)$ is a Banach space under the norm $\|f\|_p = \left(\displaystyle\int |f|^p \, d\mu\right)^{1/p}$.

In particular, $L^2$ is a Hilbert space.

Reading Corner

  • Slicing Horizontally — Lebesgue's Way of Counting [Reading] — Riemann sliced the domain vertically; Lebesgue sliced the range horizontally. Told through the metaphor of counting coins, why that single move gave birth to the convergence theorems and became the foundation of modern analysis.
  • Rebuilding "Size" — The Invention of Measure [Reading] — Extending length and area even to scattered sets, measure was a reinvention of "size" itself. From the core idea of countable additivity to the Banach–Tarski paradox that not every set can be assigned a size.

Frequently Asked Questions

What is the difference between the Lebesgue and Riemann integrals?

The Riemann integral partitions the domain vertically and sums the areas of thin strips, whereas the Lebesgue integral partitions the range horizontally and sums the measure of the set of points whose function value falls in a given band. Because of this it can integrate functions such as the Dirichlet function, whose discontinuities are dense and which is not Riemann integrable, and the monotone and dominated convergence theorems justify interchanging integral and limit for a broad class of functions. For a Riemann-integrable function on a bounded interval, the two integrals give the same value.

Why is measure theory needed?

To assign a notion of size — length, area, probability — consistently not only to intervals and rectangles but also to complicated, scattered sets, one needs the framework of $\sigma$-algebras and measures. Only once a measure is in place can the notion of "almost everywhere" be defined, which makes possible the flexible Lebesgue integral that ignores differences on null sets, as well as the rigorous foundations of probability theory and functional analysis.

What are $L^p$ spaces?

$L^p(\mu)$ is the space of measurable functions with $\int |f|^p\,d\mu < \infty$, where functions that differ only on a null set are identified, equipped with the norm $\|f\|_p = \left(\int |f|^p\,d\mu\right)^{1/p}$. This space is complete (a Banach space), and in particular $L^2$, which carries an inner product, is a Hilbert space that serves as the stage for Fourier analysis and quantum mechanics. Hölder's inequality and Minkowski's inequality are the basic tools.