Mathematica Eterna
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Abstract

Extension of Measure without Recourse to Outer Measure

Andriy Yurachkivsky

Let µ be a measure on a cofinal monotonically dense subring R of a Boolean δ-ring D. Denote by Rց and Rր the classes of those A ∈ D which are the greatest lower (respectively: least upper) bound of some decreasing (respectively: increasing) sequence in R. First we extend µ to these classes by monotonic continuity and then introduce the functions µ∗(A) = sup B∈Rց, B≤A µ(B) and µ ∗ (A) = inf B∈Rր, B≥A µ(B) on D. Denote A = {A ∈ D : µ∗(A) = µ ∗ (A)}. For A ∈ A we set µ(A) = µ∗(A), or, equivalently, µ(A) = µ ∗ (A). It is shown that A = D and thus extended function µ is a measure on D.