  Under mild assumptions the equivalence of the mixed Poisson process with
mixing parameter a real-valued random variable to the one with mixing
distribution as well as to the mixed Poisson process in the sense of Huang is
obtained, and a characterization of each one of the above mixed Poisson
processes in terms of disintegrations is provided. Moreover, some examples of
"canonical" probability spaces admitting counting processes satisfying the
equivalence of all above statements are given. Finally, it is shown that our
assumptions are essential for the characterization of mixed Poisson processes
in terms of disintegrations.
