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][1] 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][2] 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. 

  [1]: http://math.stackexchange.com/questions/70689/total-variation-distance-and-mutually-singular-probability-measure
  [2]: http://math.stackexchange.com/questions/1923914/references-about-distances-between-singular-probability-measures