On the functional derivative with respect to the Dirac Delta

It is a well-known fact that the distribution of the Bayes posterior point process is given by a ratio of functional derivatives (with respect to the Dirac delta) of a joint target-measurement probability generating functional. In the literature, functional derivatives with respect to the Dirac delta are often described as a heuristic. In this paper an existing definition of the functional derivative with respect to the Dirac delta using approximate identities is proven to be mathematically correct under general assumptions that are valid for almost all practical multitarget tracking applications.