[[Analysis MOC]]
# 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 set|Borel sets]]
- A [[measurable function]] is a homomorphism of measurable spaces.
- [[Almost everywhere]] with respect to a measure.

## Integration

- [[Lebesgue integral]]
- [[Radon-Nikodym theorem]]

## Different measures on a space

- Types of measure
  - [[Locally finite measure]]
  - [[Regular measure]]
- Particular measures
  - [[Trivial measure]] gives all sets zero mass.
  - [[Haar measure]] of a topological group.

## Categories

- [[Category of measurable spaces]] $\cat{Mble}$
- [[Category of enhanced measurable spaces]] $\cat{EMble}$
- [[Category of measure spaces]] $\cat{Mure}$

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