*Accepted Paper*

**Inserted:** 15 jan 2014

**Last Updated:** 10 dec 2017

**Journal:** Proc. Royal Soc. Edinburgh Ser. A - Mathematics

**Year:** 2016

**Abstract:**

We show that for a large class of measurable vector fields in the sense of N. Weaver (i.e. derivations over the algebra of Lipschitz functions), the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ``flows along'' the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

**Keywords:**
measurable vector field, continuity equation, flow of measures, integral curve

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