Uniform strong law on a compact parameter set
= Uniform strong law on a compact parameter set
If $\Theta$ is <compact>, $q(\theta,x)$ is continuous in $\theta$ and measurable in $x$, and $\mathbb E\sup_\Theta|q(\theta,X)|<\infty$, then an <independent and identically distributed> sample satisfies $\sup_\Theta|P_nq_\theta-Pq_\theta|\to0$ almost surely. <Uniform continuity> for each $x$, the <dominated convergence theorem>, and finite parameter nets produce arbitrarily narrow <function brackets>.