My question arises from the attached post . In this post, it is mentioned that no convergence (not even a weak one) of indicator functions can be deduced from the convergence of the sets in the sense of Hausdorff distance.
However, I wonder the following. If we could impose some extra condition ( maybe on the sets), then would the convergence of the corresponding indicator functions be true?
I already tried to google this but nothing useful showed up, so I thought to ask here.
Many thanks in advance!