  We study the dynamics of stochastic families of rational maps on the
projective line. As such families can be infinite and may not typically be
defined over a single number field, we introduce the concept of generalized
adelic measures, generalizing previous notions introduced by Favre and
Rivera-Letelier and Mavraki and Ye. Generalized adelic measures are defined
over the measure space of places of an algebraic closure of the rationals,
using a framework established by Allcock and Vaaler. This turns our heights
from sums over places into integrals. We prove an equidistribution result for
generalized adelic measures, and use this result to prove an equidistribution
result for random backwards orbits in stochastic arithmetic dynamics.
