  We study Differential Privacy in the abstract setting of Probability on
metric spaces. Numerical, categorical and functional data can be handled in a
uniform manner in this setting. We demonstrate how mechanisms based on data
sanitisation and those that rely on adding noise to query responses fit within
this framework. We prove that once the sanitisation is differentially private,
then so is the query response for any query. We show how to construct
sanitisations for high-dimensional databases using simple 1-dimensional
mechanisms. We also provide lower bounds on the expected error for
differentially private sanitisations in the general metric space setting.
Finally, we consider the question of sufficient sets for differential privacy
and show that for relaxed differential privacy, any algebra generating the
Borel $\sigma$-algebra is a sufficient set for relaxed differential privacy.
