Signed Measures and the Radon-Nikodym Theorem

Abstract

Recall that a non-empty collection Γ of subsets of the set X is called a ring of subsets of X if A ∈ Γ, B ∈ Γ implies A ∪ B ∈ Γ and A\B ∈ Γ (see Example 3.4(i)). It follows then that finite unions and finite intersections of sets in Γ are sets in Γ. The ring Γ is called an algebra of subsets of X if X itself is a member of Γ. The algebra Γ is called a σ-algebra if any countable union of sets in Γ is again a set in Γ (see Example 3.4(ii)). The mapping v from the algebra Γ into ℝ is called a (real) finitely additive signed measure (or a charge) on Γ if v(A₁ ∪ A₂) = v (A₁) + v (A₂) holds for all disjoint A_l and A₂ in Γ (see Example 4.3(7)). If Γ is a σ-algebra and v $\left( { \cup _1^\infty {A_n}} \right) = \sum\nolimits_1^\infty {v\left( {{A_n}} \right)}$holds for every disjoint sequence (A _n : n = 1, 2,…) in Γ, then v is called a σ-additive signed measure on Γ. In this section we shall briefly say “signed measure” when a σ-additive signed measure is meant. If there is only finite additivity, this will be explicitly mentioned.