$P()$ is a probability density function.
What does it mean that the density $P(v)$ is a dimensional quantity, having dimensions inverse to the dimensions of $v$?
I must be using the wrong definition of dimension. It doesn't make sense that if I have a density function over $[0,1]\times [0,1]$, the density function has dimension $\frac{1}{2}$.
I'm guessing this is related to some physics notion.
The probability is a dimensionless quantity, so since $\mathbb{P}(A) = \int P(v) dv$ if the "velocity" element has dimension $[v]$ it must be that $P(v)$ has dimension $[v]^{-1}$ in order for the probability to have no dimension.