A radon-nikodym theorem for finitely additive set functions

Suppose that Σ is a field of subsets of the set S, and suppose that μ and γ are complex-valued finitely additive set functions defined on Σ. Assume that μ is bounded and γ is finite and absolutely continuous with respect to μ. (A word of warning is in order here. The statement “γ is absolutely continuous with respect to μ” is often interpreted as “μ(E) = 0 implies γ(E) = 0”. This is not the meaning used here. Our definition is “for every ε > 0 there is a ε > 0 such that |μ(E)| < δ implies |γ(E)| < ε.” Unless μ is bounded and countably additive, the two definitions are not equivalent.) THEOREM 1. There exists a sequence {f_n} of μ-simple functions on S, such that uniformly for E∈Σ where v(μ) is the total variation of μ. Theorem 1 is established by a pure existence proof, and gives no indication of how to find f_n. A more constructive result is given below. A partition of S is a finite collection of sets E_i belonging to Σ, such that S is the disjoint union of the E_i, and such that μ(E_i) ≠ 0, i = 1,···, n. The set P of partitions may be directed by refinement, that is, by the following partial order: P₁ < P₂ if for every E∈P₁ there exist F₁,···, F_r∈P₂ (r may depend on E) such that E and differ by a μ-null set. If P is a partition of S, define the μ-simple function to be, where x_E is the characteristic function of E. THEOREM 2. If μ is positive, then uniformly for E ∈ Σ, where P is directed as explained above.