If you're going to start using probability density functions instead of just probability functions, I'd introduce their type and consider using a different symbol (e.g. lowercase p) for PDFs. I expect some people to get fairly confused when you use $~$\mathbb P$~$ but drop in a $~$\operatorname{d}\!f$~$ in out of nowhere -- "how did a tiny distance come into this!?" cries a reader-model.

Also, I'm not sure it's standard to include the delta in the density -- I'm more familiar with notation saying that the uniform PDF is 1 everywhere, and the cumulative distribution is the integral times $~$\operatorname{d}\!f$~$.