L_p spaces over finitely additive measures

Charles L. Fefferman

Vol. 26 (1968), No. 2, 265–271

DOI: 10.2140/pjm.1968.26.265

Abstract

For a space (S,Σ,μ), μ a positive finitely additive set function on a field Σ of subsets of the set S, L_p(S,Σ,μ) is usually not complete. However, if we consider the completion L_p(S,Σ,μ) of L_p, we may ask which of the properties of L_p known for the countabiy additive case, are true in general.

In this paper it is shown that for every (S,Σ,μ) there is a (countably additive) measure space (S′,Σ′,μ′) and a natural injection j from S into S′ which induces isometric isomorphisms j_∗ from L_p(S,Σ,μ) onto L_p(S′,Σ′,μ′). j_∗ also preserves order, and other structures on L_p.

This result shows, roughly, that any theorem valid for L_p over a measure space, applies also to L_p over a finitely additive measure. Thus L_p and L_q are dual (1 < p < +∞,1∕p + 1∕q = 1), L₁ is weakly complete, and so forth.

Mathematical Subject Classification

Primary: 46.35

Milestones

Received: 16 September 1966

Published: 1 August 1968