Measure

This chapter introduces the concept of a measure and other closely related notions. It starts with some examples and then introduces the concept of a σ-algebra, which is crucial in measure theory and probability theory. The σ-algebra generated by a random variable can be interpreted as the set of events that is represented by this random variable. The pair (Ω, 𝒜) consisting of a nonempty set Ω and a σ-algebra 𝒜 on Ω is called a measurable space. Such a measurable space is crucial for the definition of a measure. Next, the chapter provides some important examples of measures, including the counting measure, the Dirac measure, and the Lebesgue measure. Finally, it considers continuity and uniqueness properties of a measure. In probability theory, together with Ω and a probability measure, each σ-algebra on Ω represents a random experiment that is in some sense contained in an (often larger) random experiment.