A constructive definition of the \(n\)-dimensional \(\nu(S)\)-integral in terms of Riemann sums (Q1912725)

Introduced by Nonnenmacher and Jurkat as a descriptive integral (i.e., in terms of an additive almost everywhere differentiable set function), the \(\nu(S)\)-integral is a non-absolutely convergent integral in \(\mathbb{R}^n\) which allows to integrate the divergence of differentiable vector fields over quite general sets. The present paper proposes an equivalent definition based upon Riemann sums and proves that every function which is variationally integrable in the sense of Pfeffer is \(\nu(S)\)-integrable and that both integrals coincide.