Partial monotonicity of entropy measures

The quantification of entropy has prominence in a diverse range of fields of study including information theory, quantum mechanics, thermodynamics, ecology, evolutionary biology and even sociology. Suppose we interpret the entropy of a random object as a measurement of the uncertainty about its outcome. This measurement is expected to decrease when the object’s outcome is confined into a shrinking interval. Entropies conforming to this intuition are thus sensible and likely useful measures of uncertainty. In this paper, we give a necessary and sufficient condition for the Shannon entropy of an absolutely continuous random variable to be an increasing function of the interval. Similar results are also obtained for the Renyi entropy of absolutely continuous random variables and their convolution.