Let $E$ be a space of observables in a sequence of trials $ξ_n$ 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 $ψ$-weak topology of measures, when the sequence $ξ_n$ is assumed to be stationary. In this respect, the limit variable is naturally described as a certain canonical conditional distribution. Then, given some functional $τ$ defined on a space of laws, the consistency of the estimators $τ(m_n)$ 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 $τ$.