Integral representation without additivity

Let I I be a norm-continuous functional on the space B B of bounded Σ -measurable real valued functions on a set S S, where Σ is an algebra of subsets of S S. Define a set function v v on Σ by: v (E) v (E) equals the value of I I at the indicator function of E E. For each a a in B B let \[ J (a) = ∫ − ∞ 0 (v (a ≥ α) − v (S) ) d α