The notion of utility maximising entropy (u-entropy)of a probability density, which was introduced and studied in, is extended in two directions.First, the relative u-entropy of two probability measures in arbitrary probability spaces is defined. Then, specialising to discrete probability spaces, we also introduce the absolute u-entropy of a probability measure. Both notions are based on the idea, borrowed from mathematical finance, of maximising the expected utility of the terminal wealth of an investor. Moreover, u-entropy is also relevant in thermodynamics,as it can replace the standard Boltzmann- Shannon entropy in the Second Law. If the utility function is logarithmicor isoelastic(a powerfunction), then the well-known notions of Boltzmann-Shannon andRenyi relative entropy are recovered. We establish the principal properties of relative and discrete u- entropy and discuss the links with several related approaches in the lierature.
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