This is the follow up to this post on the completion of the space of Radon measures with respect to transport norm. In the that post we have seen that
i.e. that the completion of the Radon measure with zero total mass with respect to he dual Lipschitz norm
where is some base point in the metric space .
Recall that on a compact metric space we have , the space of Radon measures on . The Kantorovich-Rubinstein norm for measure is defined by
Theorem 1 For a compact metric space it holds that
Proof: We give an explicit identification for as follows:
- Define a Lipschitz function from an element of : For every and we set
Now we check that by linearity and continuity of that for any it holds that
This shows that is a bounded function. Similarly for all we have
This shows that is actually Lipschitz continuous, and moreover, that is continuous with norm .
- Define an element in from a Lipschitz function: We just set for and
By the definition of the -norm we have
which shows that can be extended to a continuous and linear functional , i.e. , and we also have that .
- Check that and invert each other: Finally we check that and are inverses of each other. We begin with : For and observe that
i.e. is the identity on . Conversely, for any and we check
By density of the Dirac measures in we conclude that indeed , i.e. is the identity on .