I hear that the measure of the path integral, which seems to be a measure on the space of paths is not well defined. Or that maybe the infinite dimensional measure, dx1*dx2*dx3*...*dxn is not well defined. But that's all the explanation I seem to be able to get. Most of the time "undefined" means it is infinite or has more than one value depending on how you evalute it. I'd like clarification on the meaning of "undefined" as well as you do. For it seems impossible for the path integral formulation to be fundamental if it is based on undefined measures in mathematical analysis.
Just guessing here, but maybe it has something to do with the inabilitiy to form larger measures from the measure of disjoint unions of smaller subsets of paths. Or maybe the infinite dimensions violates the requirement to have only a finite intersection of subsets. Or maybe it has something to do with the product of distributions not generally being well defined. I really don't know.