Measure

A measure $μ$ on a measurable space $(Ω, F)$ is a function:

μ : F \to [0, \infty]

satisfying:

μ (\emptyset) = 0.

Countable additivity:
For any pairwise disjoint sets $A_{1}, A_{2}, \dots \in F$ ,

μ (⋃_{i = 1}^{\infty} A_{i}) = \sum_{i = 1}^{\infty} μ (A_{i}) .

A triple $(Ω, F, μ)$ is called a measure space.

A probability measure $P$ is a measure on a measurable space $(Ω, F)$ such that

P (Ω) = 1 (the total "mass" is 1) .

A triple $(Ω, F, P)$ is called a probability space or probabilistic space.