  Let $X$ be a locally compact, second countable Hausdorff topological space.
We consider a family of commuting Hermitian operators $a(\Delta)$ indexed by
all measurable, relatively compact sets $\Delta$ in $X$ (a quantum stochastic
process over $X$). For such a family, we introduce the notion of a correlation
measure. We prove that, if the family of operators possesses a correlation
measure which satisfies some condition of growth, then there exists a point
process over $X$ having the same correlation measure. Furthermore, the
operators $a(\Delta)$ can be realized as multiplication operators in the
$L^2$-space with respect to this point process. In the proof, we utilize the
notion of $\star$-positive definiteness, proposed in [Y. G. Kondratiev and T.\
Kuna, {\it Infin. Dimens. Anal. Quantum Probab. Relat. Top.} {\bf 5} (2002),
201--233]. In particular, our result extends the criterion of existence of a
point process from that paper to the case of the topological space $X$, which
is a standard underlying space in the theory of point processes. As
applications, we discuss particle densities of the quasi-free representations
of the CAR and CCR, which lead to fermion, boson, fermion-like, and boson-like
(e.g. para-fermions and para-bosons of order 2) point processes.
  In particular, we prove that any fermion point process corresponding to a
Hermitian kernel may be derived in this way.
