Prokhorov Metric - Understanding the Definition

[peter.a]

Homework Statement

I am wondering if anyone understands why this metric is defined the way it is because i can't 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 can't 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 can't understand the next bit either which is the subsequent definition for the actual metric ie

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

Homework Equations

The Attempt at a Solution

 

[Ray Vickson]

Homework Statement

I am wondering if anyone understands why this metric is defined the way it is because i can't 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 can't 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 can't understand the next bit either which is the subsequent definition for the actual metric ie

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

Homework Equations

The Attempt at a Solution

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

RGV

 

[peter.a]

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