Measure theory MOC
Measure theory considers generalizations of length, area and volume of sets, and defines integration for exotic spaces.
Fundamentals
- A measure space consists of a measurable space (space with a σ-algebra) and a measure.
- For topological spaces, we usually deal with Borel sets
- A measurable function is a homomorphism of measurable spaces.
- Almost everywhere with respect to a measure.
Integration
Different measures on a space
- Types of measure
- Particular measures
- Trivial measure gives all sets zero mass.
- Haar measure of a topological group.