Vol. 26, No. 2, 1968

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Lp spaces over finitely additive measures

Charles L. Fefferman

Vol. 26 (1968), No. 2, 265–271

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

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

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

Mathematical Subject Classification
Primary: 46.35
Received: 16 September 1966
Published: 1 August 1968
Charles L. Fefferman
Department of Mathematics
Princeton University
Princeton NJ 08544
United States