Well the mathematicians care for those pathological cases. Probably the most important of said pathological cases is Brownian motion / Weiner processes / SDEs. Brownian motion is differential with zero probability, yet it has many modeling applications. It is also fractal-like (the self-similarity property).

The more practical advantage measure theory provides for probability is you can simultaneously handle continuous and discrete distributions. Most of the time what works for one works for the other but you can get some weird mistakes (Shannon’s differential entropy has a few issues as a measure of information not found in the discrete case because he got lazy and just replaced the sum with an integral).

A good chunk of the time I come across measure theoretic probability papers I feel like they’re making the paper a lot more complicated and messier than it needs to be, but it does serve a purpose.