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

Let b: [0, T] × ℝd → ℝd be a bounded Borel vector field, T > 0 and let µ be a non-negative Radon measure on ℝd. 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. © 2018 Independent University of Moscow.