I would be interested in references on the topic of distances between probability measures that are singular with one another and not reduced to trivial ones. For example from here we know that total variation distance is not suitable for such a problem. In the end I am looking for a structure that would be fine enough to be able to address mutually singular Wiener measures in a non trivial way (for example it is well known that the Wiener measures associate with the two process driven by a BM are mutually singulars : $X^1_t =\sigma_1.W_t$ and $X^2_t =\sigma_2.W_t$, with $X^i_0=0, i=1,2$ as soon as $\sigma_1\not=\sigma_2$ so finding a meaningful distance between those measures is a matter of interest for me).

Best regards

Note : This a re-post from MathstackExchange where no answers nor useful comments where posted, the subject might be advanced enough to be worthy of Mathoverflow. If you feel that it is not the case not a problem as long as you can post useful elements at the MSE post.

Edit : I received an interesting idea from a contributor (Ilya who is also active on this forum) in the MSE post that I (tried to) developed there and which is elaborating on one of the metric (Wasserstein-Kantorovitch) pointed out in the article by Aryeh Kontorovich.