Integral representation without additivity

Let I I be a norm-continuous functional on the space B B of bounded Σ \Sigma -measurable real valued functions on a set S S , where Σ \Sigma is an algebra of subsets of S S . Define a set function v v on Σ \Sigma 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 \[ J ( a ) = ∫ − ∞ 0 ( v ( a ≥ α ) − v ( S ) ) d α + ∫ 0 ∞ v ( a ≥ α ) d α . J(a) = \int _{ - \infty }^0 {(v (a \geq \alpha ) - v (S))d\alpha + \int _0^\infty {v (a \geq \alpha )d\alpha .} } \] Then I = J I = J on B B if and only if I ( b + c ) = I ( b ) + I ( c ) I(b + c) = I(b) + I(c) whenever ( b ( s ) − b ( t ) ) ( c ( s ) − c ( t ) ) ⩾ 0 (b(s) - b(t))(c(s) - c(t)) \geqslant 0 for all s s and t t in S S .