Calculus for the v-integral

Since F. Riesz showed in 1909 that the dual of C[0,l] is BV[0,l] (the functions of bounded variation on [0,1] withg= v[raised 1, lowered 0](g) ) via the Stieltjes integral, obtaining a representation for the functionals on BV[0,l] has been a problem of interest. In an attempt to do so, the v-integral was first introduced in [11] which characterized the dual of AC, the space of absolutely continuous functions on [0,1] with the norm given by the variation. In this paper, some calculus of the v-integral is developed. In a process of integration by parts, we obtain a characterization for the dual of the Lipschitz functions via the v-integral. A derivative to match the v-integral is defined and shown to be an inverse operator of the v-integral. Finally, a Radon-Nikodym theorem is developed and the derivative and Radon-Nikodym derivative are discussed and compared.Mathematics, Department o