Measures and Distributions

Measures A (set) function μ : K → [0,∞], with ∅ ∈ K ⊂ 2 and μ(∅) = 0 is called additive or finitely additive if A,B ∈ K, with A ∪ B ∈ K and A ∩ B = ∅ imply μ(A ∪ B) = μ(A) + μ(B). Similarly, μ is called σ-additive or countably additive if Ai ∈ K, with ⋃∞ i=1 Ai = A ∈ K and Ai ∩ Aj = ∅ for i 6= j imply μ(A) = ∑∞ i=1 μ(Ai). It is clear that if μ is σ-additive then μ is also additive, but the converse is false; e.g., take K = 2, with Ω an infinite set and μ(A) = 0 if A is finite and μ(A) = ∞ otherwise. Certainly, if K is a (σ-)ring then (σ)additivity is neat and written as: A,B ∈ K implies μ(A+B) = μ(A)+μ(B) or Ai ∈ K implies μ (∑∞ i=1 Ai ) = ∑∞ i=1 μ(Ai), plus the implicit condition μ(∅) = 0. Usually, the above properties are referred to as μ being (σ-)additivity on K. Definition 2.1. A set function μ : A → [0,∞] is called a measure if μ is σadditive and A is a σ-algebra. If μ also satisfies μ(Ω) = 1 then μ is called a probability measure or in short a probability. Thus a tern (Ω,A, μ) is called measure space if μ is a measure on the measurable space (Ω,A). Similarly, (Ω,F , P ) is called a probability space when P is a probability on the measurable space (Ω,F). Sometimes we use the name additive measure (or additive probability) to say that μ is finitely additive on an algebra A. If μ is an additive measure and A,B ∈ A with A ⊂ B then by writing A∪B = A+(BrA) we deduce μ(A) ≤ μ(B) (monotony) and μ(BrA) = μ(B)−μ(A) if μ(A) < ∞. Moreover, if Ai ∈ A, for i = 1, . . . , n then μ ( ⋃n i=1 Ai) ≤ ∑n i=1 μ(Ai) (subadditivity); and similarly with the sub σ-additivity if μ is a measure. Note that occasionally, we have to study measures defined on σ-rings instead of σ-algebras, e.g., see Halmos [14]. Perhaps the simpler example is the Dirac measure as follows: take a fix element x0 in Ω to define δ : 2 → [0, 1], δ(A) = 1A(x0), i.e., δ(A) is equal to 1 if x0 ∈ A and is equal to 0 otherwise. This gives rise to the discrete measures after using the fact that μ(A) = ∑∞ i=1 ai mi(A) is a measure if each mi is so and ai are nonnegative real numbers.