Tightness of measures

Summary

A family of probability measures on a topological space is tight if all of them live on one compact set up to a small mass. The mass left out can be made as small as one likes. Tightness rules out mass escaping to infinity. By Prokhorov's theorem, on a Polish space a family is tight if and only if every sequence in it has a weakly convergent subsequence. So proofs of convergence in distribution often have two steps: prove tightness, then identify the limit.