  The notion of utility maximising entropy (u-entropy) of a probability
density, which was introduced and studied by Slomczynski and Zastawniak (Ann.
Prob 32 (2004) 2261-2285, arXiv:math.PR/0410115 v1), 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 logarithmic or isoelastic (a power function), then the
well-known notions of the Boltzmann-Shannon and Renyi 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
literature.
