A unique characterization of the generalized Boltzmann-Gibbs-Shannon entropy

We argue that any reasonable measure D(v|m) of the dispersion (or information deficit) of an m-continuous probability measure v relative to a reference measure m should satisfy the following natural conditions: It should be formally reasonable, restrictedly continuous, invariant under density-preserving maps, quasi-subadditive and additive. We show that these conditions already determine the dispersion functionals D(v|m) uniquely up to a common positive factor and, in this way, distinguish the generalized Boltzmann-Gibbs- Shannon entropy

$\text{S(v|m)=-c∫dm}\text{dv}\text{dm}\text{ln}\text{dv}\text{dm}$