# Prokhorov metric

1. Apr 17, 2012

### peter.a

1. The problem statement, all variables and given/known data

I am wondering if anyone understands why this metric is defined the way it is because i cant seem to make sennse of it.

I get that way we use the underlying metric space to define the borel sigma field and then the set of all borel measures, but the actual definition of the metric has me confused.

As per definition we take the open epsilon neighbourhood around a point y. The point y is taken from the set in the underlying metric space ie where the random variable takes values. Then using the metric from this space we want the distance between x and y to be less then epsilon.

I cant understand why y is taken from the underlying metric space and the x is taken from a subset of the borel sigma field, ie the image space. And i cant understand the next bit either which is the subsequent definition for the actual metric ie

P(A)<Q(A(epsilon)+epsilon

2. Relevant equations

3. The attempt at a solution

2. Apr 17, 2012

### Ray Vickson

Have you looked at http://people.math.sfu.ca/~tupper/PUBLICATIONS/sde.pdf ?

RGV

3. Apr 18, 2012

### peter.a

I have already read that and most other content on the web relating to the prokhorov metric.