  Let $\Gamma$ be a countable group and let $G$ be the Schreier graph of the
free part of the Bernoulli shift of $\Gamma$ (with respect to some finite
subset $F \subseteq \Gamma$). We show that the Borel fractional chromatic
number of $G$ is equal to $1$ over the measurable independence number of $G$.
As a consequence, we asymptotically determine the Borel fractional chromatic
number of $G$ when $\Gamma$ is the free group, answering a question of Meehan.
