Although my question is on Feynman integration I briefly digress to
discuss the Gauge Integral.

Not so long ago, one of my professors drew my attention to a theory of
integration known as Gauge integration (also called Henstock-Kurzweil
integration of generalized Riemann integration). This theory was
first develped in the 50's and has many advantages of Lebesgue's
theory. It is not my purpose here to discuss this in detail so I
refer you to:

http://atlas.math.vanderbilt.edu/~schectex/ccc/gauge/

However, let me briefly describe some advantages. This integral
arises from a slight change in the definition of the usual Riemann
integral and leads to a theory with the following properties (for
simplicity let's restrict to R):

1) The integral is defined for all Lebesgue integrable functions and
gives the same results.
2) Unlike the Lebesgue integral, it is a CONDITIONAL integral and can
therefore integrate a larger class of functions.
3) The full Fundamental Theorem of Calculus holds with no assumptions
on the continuity of the derivative.

Due to the simplicity of the definitions I have never understood why
this theory is virtually unknown in so many universities.

Since this integral is conditional for a while I was wondering if this
theory has any applications to Feynman integration. I found a single
account of application of this theory to path integrals (referenced on
the site above):

Muldowney, General theory of integration in function spaces, including
Wiener and Feynman integration, 1987.

To quote from the book (p. 75):

"Various methods have since been devised for defining the function
space integral. Ito (56) takes a limit of the finite-dimensional
measures to produce a kind of measure in the function space.
Muldowney, General theory of integration in function spaces, including
Wiener and Feynman integration, 1987. Cameron and Storvick (54) put an
imaginary component in the Wiener integral and then let a real
component tend to zero. Cecile Morette DeWitt (55) and . . ., make
use of the Gaussian form of the Feynman integrand. These definitions
are generally such that the standard theorems and methods of
integration cannot be easily applied. . .
This approach [Henstocks theory] places no prior conditions on the
functions or functionals of associated elements which we may wish to
integrate. We do not require that the integrand be positive or real,
or have Gaussian form, or contain a Lebesgue-type countable additive
integrator . . .This means that the familiar methods of integration
theory can be readily used when we tackle the problem of Feynman
integration.

All the other recent accounts of path integration I have encountered
do not even refer to the existence of this theory. I would like to
know if these ideas are familiar to the mathematics/physics community.

Thank you.