Integral representation without additivity

Let J be a norm-continuous functional on the space B of bounded Σ-measurable real valued functions on a set S, where Σ is an algebra of subsets of S. Define a set function v on Σ by: v(E) equals the value of I at the indicator function of E. For each a in B let J(a)= ⁰ ∫ _-∞(v(a ≥ α) - v(S)) dα+ ⁰ ∫ _∞ v(a ≥ α) dα.Then I = J on B if and only if I(b + c) = 1(b) + 1(c) whenever (b(s)-b(t))(c(s)-c(t)) ≥ 0 for all s and t in S.