We give some definitions and basic properties of weak convergence of probability measures on metric spaces, based on [Bil99].

Let be a metric space. are probability mesures on . We write for measurable for convenience.

Definition 1is said to be weakly convergent to , denoted by , if for every .

The following theorem provides useful conditions equivalent to definition of weak convergence. A set is called -continuity set, if .

Theorem 2(The Portmanteau Theorem, [Bil99, Theorem 2.1]) These five conditions are equivalent:

- .
- for all bounded uniformly continuous .
- for all closed
- for all open .
- for all -continuity sets .

Let be the space of probability measures on . can be metrized by *Prohov distance* ([Bil99, P. 72]) if is separable and complete metric space.

Theorem 3([Bil99, Theorem 6.8]) Suppose that is separable and complete. Then

- weak convergence is equivalent to -convergence,
- is separable and complete
- is relatively compact if and only if its -closure is -compact.

Let be another metric space, and is measurable . Then is a probability measure on .

Theorem 4(Mapping theorem, [Bil99, Theorem 2.7]) Let is measurable , and . In addition, , where be the set of discontinuities of . Then .

The above mapping theorem saids, weak convergence of probability measures are preserved by mapping , if is continuous almost surely with respect to the limit measure , i.e. .

Let be a probability space, and be measurable w.r.t. . Then, is called random element. The *distribution * of is the probability measure on , which is sometimes denoted by . If is a measurable , then .

Definition 5We say a sequence converges in distribution to , denoted by , if .

Thus, if and only if for all . We call a set as an -continuity set if . Now, we can rewrite Theorem 2 in terms of random elements.

Theorem 6([Bil99, p. 26]) These five conditions are equivalent:

- .
- for all bounded uniformly continuous .
- for all closed
- for all open .
- for all -continuity sets .

Also, the Theorem 4 can be rewritten into mapping theorem of random elements: on implies on , if is continuous almost surely with respect to , i.e.