  This work considers a problem of estimating a mixing probability density $f$
in the setting of discrete mixture models. The paper consists of three parts.
  The first part focuses on the construction of an $L_1$ consistent estimator
of $f$. In particular, under the assumptions that the probability measure $\mu$
of the observation is atomic, and the map from $f$ to $\mu$ is bijective, it is
shown that there exists an estimator $f_n$ such that for every density $f$
$\lim_{n\to \infty} \mathbb{E} \left[ \int |f_n -f | \right]=0$.
  The second part discusses the implementation details. Specifically, it is
shown that the consistency for every $f$ can be attained with a computationally
feasible estimator.
  The third part, as a study case, considers a Poisson mixture model. In
particular, it is shown that in the Poisson noise setting, the bijection
condition holds and, hence, estimation can be performed consistently for every
$f$.
