A radon-nikodym theorem for finitely additive set functions

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Abstract

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 {fn} 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 fn. A more constructive result is given below. A partition of S is a finite collection of sets Ei belonging to Σ, such that S is the disjoint union of the Ei, and such that μ(Ei) ≠ 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,···, Fr∈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 xE is the characteristic function of E. THEOREM 2. If μ is positive, then uniformly for E ∈ Σ, where P is directed as explained above.

Original languageEnglish (US)
Pages (from-to)35-45
Number of pages11
JournalPacific Journal of Mathematics
Volume23
Issue number1
DOIs
StatePublished - Oct 1967

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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