Introduction to Measure Theory

Introduction to Measure Theory

Motivation

-Algebra

Measure

Measure Space

is a measure space.

Lebesgue Measure

The Lebesgue measure extends length to a measure on a -algebra containing all "reasonable" subsets of .

Borel Sets

Lebesgue measurable sets

Non-Measurable Sets

Simple Functions

Lebesgue Integral

Monotone Convergence Theorem

Fatou's Lemma

Dominated Convergence Theorem

Comparison with Riemann Integral

Spaces

Hölder's Inequality

Minkowski's Inequality