On Measure Theoretic definitions of Generalized Information Measures and Maximum Entropy Prescriptions
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Though Shannon entropy of a probability measure $P$, defined as $- \int_{X} \frac{\ud P}{\ud μ} \ln \frac{\ud P}{\udμ} \ud μ$ on a measure space $(X, \mathfrak{M},μ)$, does not qualify itself as an information measure (it is not a natural extension of the discrete case), maximum entropy (ME) prescriptions in the measure-theoretic case are consistent with that of discrete case. In this paper, we study the measure-theoretic definitions of generalized information measures 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 definitions of generalized relative-entropies, Rényi and Tsallis, are natural extensions of their respective discrete cases, (ii) we show that, ME prescriptions of measure-theoretic Tsallis entropy are consistent with the discrete case.