Effective notions of weak convergence of measures on the real line

Abstract: We establish a framework for the study of the effective theory of weak convergence of measures. We define two effective notions of weak convergence of measures on $\mathbb{R}$: one uniform and one non-uniform. We show that these notions are equivalent. By means of this equivalence, we prove an effective version of the Portmanteau Theorem, which consists of multiple equivalent definitions of weak convergence of measures.