  We provide here a novel algebraic characterization of two information
measures associated with a vector-valued random variable, its differential
entropy and the dimension of the underlying space, purely based on their
recursive properties (the chain rule and the nullity-rank theorem,
respectively). More precisely, we compute the information cohomology of Baudot
and Bennequin with coefficients in a module of continuous probabilistic
functionals over a category that mixes discrete observables and continuous
vector-valued observables, characterizing completely the 1-cocycles; evaluated
on continuous laws, these cocycles are linear combinations of the differential
entropy and the dimension.
