For some analysis work with probability distributions, I remember a common trick being to drop the "integrate to 1" requirement, so the set becomes closed under addition and is more convenient to work with in general. It's always easy to re-normalize at the end, if needed.
I cannot for the life of me remember what these more general objects are called, though. Are they just "distributions" or is there a more specific name? Again, the name for the members of a function space with all the properties of a PDF except the requirement that they integrate to 1.