Definition 1   Let $\M=\{\mu_i,i \in I \}$ be a family of finite measures on the Borel subsets of a metric space $\Omega$ . We say that $\M$ is tight iff for each $\epsilon>0$ there is a compact set $K$ such that $\mu_i(\Omega-K)<\epsilon$ for all $i$ . We say that $\M$ is relatively compact iff each sequence in $\M$ has a subsequence converging weakly to a finite measure on $\B(\Omega)$ .

If $\{F_i,i \in I \}$ is a family of distribution functions, relative compactness or tightness of $\{F_i\}$ refers to relative compactness or tightness of the corresponding measures.

Theorem 1   Let $\{F_i,i \in I \}$ be a family of distribution functions with $F_i(\infty)-F_i(-\infty)<M<\infty$ for all $i$ . The family is tight iff it is relatively compact.

Proof. Coming soon...(needs other theorems before)