L_p Spaces over finitely additive measures

For a space a positive finitely additive set function on a field Σ of subsets ofthe se is usually not complete. However, if we consider the completion we may ask which of the properties of Lp known for the countably additive case, are true in general.In this paper it is shown that for every there is a (countably additive) measure space and a natural injection j from S into Sr which induces isometric isomorphisms y*from 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 L_p over a finitely additive measure. Thus L_p and L_q are dual L_t is weakly complete, and so forth.