  In this article, we show that every stationary random measure on $\mathbb
R^d$ that is essentially free (i.e., has no symmetries a.s.) admits a point
process as a factor (i.e., as a measurable and translation-equivariant function
of the measure). As a result, we improve the results of Last and Thorisson
(2022) on the existence of a factor balancing allocation between ergodic pairs
of stationary random measures $\Phi$ and $\Psi$ with equal intensities. In
particular, we prove that such an allocation exists if $\Phi$ is diffuse and
either $(\Phi,\Psi)$ is essentially free or $\Phi$ assigns zero measure to
every $(d-1)$-dimensional affine hyperplane. The main result is deduced from an
existing result in descriptive set theory, that is, the existence of lacunary
sections. We also weaken the assumption of being essentially free to the case
where a discrete group of symmetries is allowed.
