  The thermodynamic maximum principle for the Boltzmann-Gibbs-Shannon (BGS)
entropy is reconsidered by combining elements from group and measure theory.
Our analysis starts by noting that the BGS entropy is a special case of
relative entropy. The latter characterizes probability distributions with
respect to a pre-specified reference measure. To identify the canonical BGS
entropy with a relative entropy is appealing for two reasons: (i) the maximum
entropy principle assumes a coordinate invariant form; (ii) thermodynamic
equilibrium distributions, which are obtained as solutions of the maximum
entropy problem, may be characterized in terms of the transformation properties
of the underlying reference measure (e.g., invariance under group
transformations). As examples, we analyze two frequently considered candidates
for the one-particle equilibrium velocity distribution of an ideal gas of
relativistic particles. It becomes evident that the standard J\"uttner
distribution is related to the (additive) translation group on momentum space.
Alternatively, imposing Lorentz invariance of the reference measure leads to a
so-called modified J\"uttner function, which differs from the standard
J\"uttner distribution by a prefactor, proportional to the inverse particle
energy.
