On regular extensions of measures

George Bachman and Alan Sultan

Vol. 86 (1980), No. 2, 389–395

DOI: 10.2140/pjm.1980.86.389

Abstract

Let ℒ be a sublattice of 2^X containing ∅, X and let MR(ℒ) be the collection of all bounded nonnegative finitely additive measures defined on 𝒜(ℒ) the algebra generated by ℒ which are ℒ-regular in the sense that μ(E) = supμ(L), L ⊂ E,L ∈ℒ,E ∈𝒜(ℒ). It is shown here that if ℒ₁ ⊂ℒ₂ are sublattices of 2^X and μ ∈ MR(ℒ₁), then μ extends to a ν ∈ MR(ℒ₂). Several applications are given.

Mathematical Subject Classification 2000

Primary: 28C15

Secondary: 54H99

Milestones

Received: 6 April 1979

Revised: 21 June 1979

Published: 1 February 1980