Abstract
This chapter discusses the permanence of operator-theoretic properties of vector measures given by Riesz theorems under three separate notions of absolute continuity. It discusses whether compactness and weak compactness of operators are preserved under various absolute continuity conditions. A representing measure is countably additive if and only if it is regular. If each of n and m is a representing measure, then n is absolutely continuous with respect to m(n < m) if for each ∈ > 0, there is a δ > 0. The reflexivity of E immediately implies that n(A) is a weakly compact operator for each A ∈ Σ.
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