Assume $V$ has only log terminal singularities. A positive definite adapted measure on $V$ is a positive Radon measure locally of the form $e^{f}.v$ where $f$ is a bounded measurable function. A positive definite adapted measure has ${\mathcal{C}}^{0}$, ${\mathcal{C}}^{\alpha}$, ${\mathcal{C}}^{\infty}$ density if so is $f$.