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

Let b: [0, T] × ℝ^d → ℝ^d be a bounded Borel vector field, T > 0 and let µ be a non-negative Radon measure on ℝ^d. We prove that a µ-measurable flow of b exists if and only if the corresponding continuity equation has a non-negative measure-valued solution with the initial condition µ.