On the f-divergence for discrete non-additive measures

Abstract In this paper we study the definition of the f-divergence and the Hellinger distance for non-additive measures in the discrete case. As these measures are based on the derivatives of the measures, we consider the problem of defining the Radon–Nikodym derivative of a non-additive measure. While Radon–Nikodym derivatives for additive measures exist for absolutely continuous measures, this is not the case in the non-additive case. In this paper we will define set-directional and upper, lower and interval derivatives. We will also define when two measures have the same sign. These definitions will be used to introduce alternative definitions of the f-divergence, all extending the classical definition to non-additive measures.

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