It matters by convention, because the textbooks are written that way.

My point is that you don't need that level of formal rigour to do applied work. You can derive the Feynman-kac formula via a scaling limit of discrete-time Markov chains. Add some levy process (a.k.a compound Poisson processes) and you're basically done.

If you want to be ultra-rigourous in your definitions, then you need measure theory, yes. But even Einstein didn't need that for his description of Brownian motion. If a scaling limit is good enough for him, it's good enough for me.