= 1

F( , , ) = X ( , L). Theorem 6. Let X : × R × Ł R+ be a positive, increasing, right continuous, adapted, integrable, 1-process measure. Then X is 1-summable relative to the embedding R L(R,R). The measure IX is —additive and has finite variation |IX|. Proof: Let F : , × R × L R+ be the function associated with X by the previous theorem. Since X is integrable we have F(•, , ) = X (•, L) L1. By theorem 2.5 there is a positive, finite, —additive measure F : F B(R) ,Ł R+ satisfying

F(C) = E( 1CdFz), for C F B(R) Ł. For C in the ring generated by the sets A × (t, t'] × (x, x'] with A Ft,t t' in R and x x' in L, we have

F(C) = E 1F(C) = E Ix(C) = ||IX(C)||L1

It follows that IX has finite variation |IX| and |IX| = F on the ring generated by the above sets.