  Radon-Nikodym (RN) derivative between two measures arises naturally in the
affine structure of the space of probability measures with densities. Entropy,
free energy, relative entropy, and entropy production as mathematical concepts
associated with RN derivatives are introduced. We identify a simple equation
that connects two measures with densities as a possible mathematical basis of
the entropy balance equation that is central in nonequilibrium thermodynamics.
Application of this formalism to Gibbsian canonical distribution yields many
results in classical thermomechanics. An affine structure based on the
canonical represenation and two divergences are introduced in the space of
probability measures. It is shown that thermodynamic work, as a conditional
expectation, is indictive of the RN derivative between two energy
represenations being singular. The entropy divergence and the heat divergence
yield respectively a Massieu-Planck potential based and a generalized Carnot
inequalities.
