Integral representation without additivity

Let I I be a norm-continuous functional on the space B B of bounded Σ Σ -measurable real valued functions on a set S S , where Σ Σ is an algebra of subsets of S S . Define a set function v v on Σ 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