Functional approach of large deviations in general spaces (Q1776121)

Let \((\mu_{\alpha})\) be a net of Borel probability measures on a topological space \(X\), and \((t_{\alpha})\) a net in \(]0,{\infty}[\) converging to \(0\). For each \([-{\infty}, +{\infty}[\)-valued measurable function \(h\) on \(X\), we define \(\Lambda(h)=\log \lim (\int_X \exp{{h(x)}\over{t_\alpha}} \mu_{\alpha}(dx))^{t_\alpha}\) provided the limit exists in \([-\infty,+\infty]\). In order to prove a large deviation principle for \((\mu_{\alpha}^{t_\alpha})\), the so-called functional approach consists first to establish the existance of \(\Lambda(\cdot)\) on some class of continuous functions, and next to look for a sufficient condition, either of set-theoretic type, or on the functional \(\Lambda(\cdot)\). The author deals only with set-theoretic conditions. He presents a Prokhorov-type theorem with a weaker notion than exponential tightness, which improves known results.