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 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
• Category of enhanced measurable spaces
• Category of measure spaces

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