Notes on probability measures on metric spaces

1 Borel sets

2 Borel probability measure

3 Weak convergence of measures

4 The Prohkorov metic

Let (X,d) be a metric space. Denoted by P(X) all the Borel probability measure on X.

We have defined the notation  of weak convergence in P(X), Define for \mu,\nu\in P(X)

\displaystyle d_P(\mu,\nu):=\inf\{\alpha>0:\mu(A)\leq \nu(A_\alpha)+\alpha\ \text{and}\ \nu(A)\leq \mu(A_\alpha)+\alpha\}

where

A_\alpha=\{x:d(x,A)<\alpha\}.

The function d_P is called the Prokhorov metric on P(X) (induced by  d).

Conclusion If (X,d) is a separable metric space, then so is P(X) with the induced Prokhorov metric. Moreover,  a sequence in P(X) converges in metric if and only if it converges weakly and to the same limit.

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