  The main result of the paper is classification of free multidimensional Borel
flows up to Lebesgue Orbit Equivalence, by which we understand an orbit
equivalence that preserves the Lebesgue measure on each orbit. Two non smooth
Euclidean flows are shown to be Lebesgue Orbit Equivalence if and only if they
admit the same number of invariant ergodic probability measures.
