Let Ω={1,…,n} and P={X:S⊆Ω}. A mapping e : P→R + is a convex set function if e(⊖)=0 and e( S) + e( T)⩽ e( S∩ T) + e( S ∪ T) for all S. TϵP. The set of convex set functions for fixed Ω is a convex cone and the paper is dealing with the extreme points of the base ▪ 1 = {e: e(Ω)=1} of this cone. To this end a representation theorem is proved: every e ϵ ▪ 1 can be written as e(·)=max( m 1(·)− α 1… m t (·)− α t ), where m 1,…, m t are measures on P and α 1,…,α t are nonnegative reals. Given additional requirements, the representation is unique and called “ canonical”. Fix H ⊆{1,…, r},| H| ⩾ 2. There is a certain subsystem of sets SϵP such that m τ ( S)− α τ = e( S) ( τϵ H}, that is, the subsystem of sets S such that m τ ( S)− α τ ( τϵH) is a maximal term in the representation of e by m 1,…, m τ and α 1,…α t . e is called nondegenerate is these subsystems determine the measures m 1,…, m τ uniquely and it turns out that nondegeneracy and extremality are equivalent for e ϵ ▪ 1. Moreover, it is seen that nondegeneracy is closely related to a generalized version of the problem “represent a given integer λ ⩾ o by means of integer weights g,…, g r ⩾ 0 via σ r ϱ = 1 a ϱ g ϱ = λ such that the integer coefficients a ϱ satisfy 0⩽ a ϱ ⩽ k ϱ ( ϱ=1,…, r), where k ϱ are prescribed integer bounds. Find r such representations with the additional property that the coefficients form a nonsingular matrix.” A solution to the generalized version of this number theoretical problem is given and, finally, a few examples are discussed.