About Some Monge–Kantovorich Type Norm and Their Applications to the Theory of Fractals

Abstract

If X is a Hilbert space, one can consider the space cabv(X) of X valued measures defined on the Borel sets of a compact metric space, having a bounded variation. On this vector measures space was already introduced a Monge–Kantorovich type norm. Our first goal was to introduce a Monge–Kantorovich type norm on cabv(X), where X is a Banach space, but not necessarily a Hilbert space. Thus, we introduced here the Monge–Kantorovich type norm on cabvLq([0,1]),(1<q<∞). We obtained some properties of this norm and provided some examples. Then, we used the Monge–Kantorovich norm on cabvKn(K being R or C) to obtain convergence properties for sequences of fractal sets and fractal vector measures associated to a sequence of iterated function systems.