Can we refine the paper "Integration with Filters" by Bottazzi E. and Eskew M. to find a mathematically rigorous definition of the path integral?

(My parents will only let me spend $\$$100 this month. I can offer $\$$15 monthly until you get the money you need.)

Motivation:

In a magazine article on problems and progress in quantum field theory, Wood writes of Feynman path integrals, “No known mathematical procedure can meaningfully average an infinite number of objects covering an infinite expanse of space in general. The path integral is more of a physics philosophy than an exact mathematical recipe.”

This article (and its final version) provides a method for averaging an arbitrary collection of objects; however, the average can be any value in a proper extension of the range of these objects. (Notice, an arbitrary collection of these objects is a set of functions.)

As a amateur mathematician, I know nothing about path integrals. For example, I incorrectly assumed the path integral averages a function rather than a set of functions: i.e., when reading Wood’s quote in the beginning of the arXiv version of "Integration with Filters", I assumed he wanted to average a function whose graph contains “an infinite number of objects covering an infinite expanse of space”.

Question: Can we refine the final version of "Integration with Filters" to get a unique average of a set of functions, which could be used to find a mathematically rigorous definition of the path integral?