A necessary and sufficient condition for existence of measurable flow of a bounded Borel vector field

Let $b\colon[0,T]\times\mathbb R^d\to\mathbb R^d$ be a bounded Borel vector field, $T>0$ and let $\bar\mu$ be a non-negative Radon measure on $\mathbb R^d$. We prove that a $\bar\mu$-measurable flow of $b$ exists if and only if the corresponding continuity equation has a non-negative measure-valued solution with the initial condition $\bar\mu$.