When physicists talk math concepts, e.g., probability densities, differentiability, continuity, probability, inner products, Fourier transforms, solutions to differential equations, unitary transformations, etc. they are talking math. They should get the math correct.

So, roughly the paradigm might be -- as a student I assumed it was -- start with a physics problem, convert to a math problem, see what the math says, e.g., solution to a differential equation, Stokes theorem, the weak law of large numbers, then convert back to physics to see what those math results say about the physics.

The idea of a finite universe, say, with a boundary, is popular in popular science and maybe science fiction but is missing from nearly all serious, accepted physics as taught today. So, really, the idea of a finite universe is no good as a patch up of what Feynman wrote.

A simple approach for some of this issue is just to say,

"I have a lab. A neutron is wandering around in it. I don't know where it is. So, I assume the probability density of its location is uniform in the lab."

That's fine, that is, in the math.

Notice, I never used rigorous. When physics profs used that word, it meant that it was time for me, no delay, don't wait for the end of the class period, just to stand, say nothing, walk out, drop the course, and return to the math department.