I am trying to understand the topologies of the Skorohod space of càdlàg functions. I found the following sentence in one paper.
The supremum norm converts the Skorokhod space into a nonseparable Banach space, what is always disadvantageous in probability theory.
As far as I understand, the nonseparability of the Skorokhod space equipped with supremum norm is one of the main reasons to investigate different topologies of this space (for example, the metric separable topology $J_1$ introduced by Skorokhod).
My questions are as follow: why is a nonseparable Banach space always disadvantageous in probability theory; what kind of problems do we have to deal with when we have a nonseparable Banach space? Some examples or references would be really great.
Any help is much appreciated!