math414

Measure Theory

Course content

Chapter I: Lebesgue Measure

Introduction

The length of intervals

The length of open sets

Algebra and σ-algebra of sets

Countably additive measure

Outer Measure

Definition of Lebesgue outer measure

Properties and theorems on Lebesgue outer measure

Measurable Sets and Lebesgue Measure
Definition of Measurable Sets
Properties and theorems on measurable sets
Intervals and Borel sets
Definition of Lebesgue measure
Properties and theorems on Lebesgue measure

Measurable Functions
Definition of measurable functions
Properties and theorems on measurable functions
The characteristic function
Properties and theorems on simple functions
Theorems on sequences of measurable functions

Chapter II: The Lebesgue Integral

The Riemann Integral
Definition of Riemann integral
Step functions
Examples of functions that are not Riemann integrable

The Lebesgue Integral of a Bouded Function Over a Set of Finite Measure
The integral of simple functions that vanish outside a set of finite measure
Definition of the integral of a bounded function over a set of finite measure
The equivalence between measurability and integrability
The relationship between Riemann and Lebesgue integrals
Properties and theorems
The Bounded Convergence Theorem

The Lebesgue Integral of Nonnegative Functions
Definition of the Lebesgue integral of nonnegative functions
Properties and theorems
Fatou's Lemma
Monotone Convergence Theorem
Definition and theorems on integrable functions

The General Lebesgue Integral
The positive and negative parts of a function, properties and theorems
Definition of an integralble function and its integral
Properties and theorems
Lebesgue Convergence Theorem
Applications