In applied sciences, in particular in economics (with which I’m most familiar), the Cantor distribution is mentioned to show that the results obtained are very general as they hold even assuming such a ‘pathological’ case.
See e.g.
> Machina, Mark, and John Pratt. "Increasing risk: some direct constructions." Journal of Risk and Uncertainty 14.2 (1997): 103-127. [Link][1] 

When we model economic (or other real-world) situations we want the output of a model to be robust to changes in underlying assumptions. Thus, obtaining the result under less restrictive assumptions is seen as a progress in applied sciences.


  [1]: http://econweb.ucsd.edu/~mmachina/papers/Machina-Pratt_Increasing_Risk.pdf