To what extent can we call a Geometric Distribution a Geometric Density In some papers, for example in "The Geometric Density with Unknown Location Parameter" by Klotz, a Geometric Distribution is called a Geometric Density.
For me, this claim looks erroneous, however Klotz is a serious statistician and a professor in the field.
My question is, to what extend is it legitimate to call a Geometric Distribution a Geometric Density?
 A: Heuristic answer: Without much mathematic you could say that a continuous variable has a density with respect to the Lebesgue measure, and a discrete random variable has a density with respect to the counting measure.
Developped answer: The concept of density is much wider than you may think. A density of a probability measure $P$ can be defined with respect to a measure $\lambda$ that dominates $P$ by the Radon Nikodym Theorem (see http://en.wikipedia.org/wiki/Radon%E2%80%93Nikodym_theorem). Here density should be understood as a density with respect to the counting measure defined on the mentionned countable set. I agree that it is not extremely rigorous not to mention the reference when talking about a density (but who mention density  wrt lesbesgue measure?), but it pose no problem while reading the paper in question so .... 

Additional Annex Notes
 I have seen a certain number of machine learning notes (I won't do delation) where  the reference measure is not the counting measure and we see things such as $P(X=x|Y=y)$ with X being a continuous variable (with a density wrt Lebesgues) (to apply Bayes principle and derive the Bayes rule). I guess people want to be pedagogic and do not want to bother students with technical details ;) My conclusion would be that even great mathematician can commit abuse of wording or of notation (it is not the case in the paper you mention because the authors of a paper in AoP may have mathematical background), it is not a problem as along as it is understood by everyone...  
