  We prove a multidimensional ergodic theorem with weighted averages for the
action of the group $\mathbb{Z}^d$ on a probability space. At level $n$ weights
are of the form $n^{-d} \psi(j/n)$, $ j\in \mathbb{Z}^d$, for real functions
$\psi$ decaying suitably fast. We discuss applications to random measures and
to quenched stochastic homogenization of random walks (RWs) on simple point
processes with long-range random jump rates, allowing to remove the technical
Assumption (A9) from \cite[Theorem~4.4]{Fhom1}. This last result concerns also
some semigroup and resolvent convergence particularly relevant for the
derivation of the quenched hydrodynamic limit of interacting particle systems
(IPSs) via homogenization and duality. As a consequence we show that also the
quenched hydrodynamic limit of the symmetric simple exclusion process on point
processes stated in \cite[Theorem~4.1]{F_SEP} remains valid when removing the
above mentioned Assumption (A9).
