  Let $E$ be a space of observables in a sequence of trials $\xi_n$ and define
$m_n$ to be the empirical distributions of the outcomes. We discuss the almost
sure convergence of the sequence $m_n$ in terms of the $\psi$-weak topology of
measures, when the sequence $\xi_n$ is assumed to be stationary. In this
respect, the limit variable is naturally described as a certain canonical
conditional distribution. Then, given some functional $\tau$ defined on a space
of laws, the consistency of the estimators $\tau(m_n)$ is investigated. Hence,
a criterion for a refined notion of robustness, that applies when considering
random measures, is provided in terms of the modulus of continuity of $\tau$.
