Quasispaces induced by vector fields measurable in $\mathbb R^3$

We study some metric functions that are induced by a class of basis vector fields in $\mathbb R^3$ with measurable coordinates. These functions are proved to be quasimetrics in the domain of definition of the vector fields. Under some natural constraints, the Rashevsky–Chow Theorem and the Ball-Box Theorem are established for the classes of vector fields we consider.