On measure-theoretic aspects of nonextensive entropy functionals and corresponding maximum entropy prescriptions

Shannon entropy of a probability measure P, defined as -∫X(dP/dμ)ln(dP/dμ)dμ on a measure space (X,M,μ), is not a natural extension from the discrete case. However, maximum entropy (ME) prescriptions of Shannon entropy functional in the measure-theoretic case are consistent with those for the discrete case. Also it is well known that Kullback–Leibler relative entropy can be extended naturally to measure-theoretic case. In this paper, we study the measure-theoretic aspects of nonextensive (Tsallis) entropy functionals and discuss the ME prescriptions. We present two results in this regard: (i) we prove that, as in the case of classical relative-entropy, the measure-theoretic definition of Tsallis relative-entropy is a natural extension of its discrete case, and (ii) we show that ME-prescriptions of measure-theoretic Tsallis entropy are consistent with the discrete case with respect to a particular instance of ME.