Abstract
Let ( Ω, τ, m) be a finite, nonatomic, separable measure space. This paper extends the Radon-Nikodym theorem to odd, disjointly additive, m-continuous functionals whose domain consists of all differences of characteristic functions which belong to a given subspace of L ∞( m). Such a functional will possess a density in L 1( m) provided that the subspace is weak ∗-closed and separates sets; the conclusion can fail if the latter hypothesis is removed. Analogous results are obtained for functionals which are not necessarily odd.
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