On Measure Theoretic definitions of Generalized Information Measures and Maximum Entropy Prescriptions
<|reference_start|>On Measure Theoretic definitions of Generalized Information Measures and Maximum Entropy Prescriptions: Though Shannon entropy of a probability measure $P$, defined as $- \int_{X} \frac{\ud P}{\ud \mu} \ln \frac{\ud P}{\ud\mu} \ud \mu$ on a measure space $(X, \mathfrak{M},\mu)$, 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, R\'{e}nyi 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.<|reference_end|>
@article{dukkipati2006on,
    title={On Measure Theoretic definitions of Generalized Information Measures and
  Maximum Entropy Prescriptions},
    author={Ambedkar Dukkipati, M Narasimha Murty and Shalabh Bhatnagar},
    journal={arXiv preprint arXiv:cs/0601080},
    year={2006},
    archivePrefix={arXiv},
    eprint={cs/0601080},
    primaryClass={cs.IT math.IT}
}