Assume $V$ has only log terminal singularities. An adapted measure on $V$ is a positive Radon measure locally of the form $e^{f}.v$ where $f$ is locally given as the sum of a psh and a smooth function on $V$. An adapted measure is ${\mathcal{C}}^{0}$, ${\mathcal{C}}^{\alpha}$, ${\mathcal{C}}^{\infty}$ density if so is $e^{f}$.