We already have logdensity(d::Distribution, x) == logpdf(d, x). But we want to keep them separate, for a couple of reasons.
First, as you mention, only some measures are also probability distributions, so logdensity is more fitting as a general term.
Second, we want to have logdensity(d, base, x) where base is taken as the base measure (assuming d is absolutely continuous relative to it).
Yes, you could create a redundantDirichlet. I don’t know the internals of Turing very well, but I’d guess this would be one of the easier steps in getting it to work with MeasureTheory. But I hope I’m wrong about that, since it would be great to have Turing easily set up to use it 