@Rene I have added some informal comments and another reference. Hope this helps.

What is the limit of $B_\epsilon$ as $\epsilon \to 0$ if $b$ is $BV$?

Ok thanks for providing a reasonably understandable definition. :) So a cardinal $\alpha$ is an Ulam number if - roughly speaking - every set of cardinality less then $\alpha$ has measure $0$ for (a class of outer) measures $\mu$. Fair; but what does it mean that the minimum cardinality of a dense set is an Ulam number? Is it like saying (I am thinking to $\mathbb R^N$) that countable (=cardinality of dense subset, being euclidean spaces separable) sets are negligible for every non-atomic (outer) measure?

@PiotrHajlasz I am sorry to bother you but, if you have some time, I would read with pleasure your comment/answer (especially concerning the link between BD functions and currents). Thanks a lot for your help.

@PaataIvanishvili Nice observation about the semi-circle law! I now see the analogy with S. Lee's answer. I did want to try a similar approach but I thought that since we do not have any longer a "scalar product" that was not meaningful. Probably the correct way of writing it is by observing that some integrals has to be independent of the other variables... I'm going to try in a moment and I'll update if I have some news. Thanks!

@Dan Thanks for your comments. I appreciate your contribution and I do not find it useless. Indeed, I decided to award the bounty to you, as you provided me a lot of new tools and theorem to think about. I hope the others who provided an answer do not mind: I acknowledge them too, of course, but your answer is more in the spirit of what I was looking for.

Gosh, thanks a lot for such an answer! I am very happy to read your contribution. I am not an expert in probability (I am not familiar with De Finetti or Diaconis-Friedman theorem) so I need sometime to study and understand your approach, which looks interesting. As a "concrete" example, may I ask you to kindly give a look here (where actually the whole story began)? Probably in that specific example your approach can yield the final answer... Thanks again!