In complex analysis one frequently applies results from Lebesgue's theory of integrals to path integrals. For example interchanging the order of integration in double path-integrals by refering to Fubini's Theorem.
This always bothers me when it is done without any further comment because Fubini's Theorem applies to integrals on measure spaces. I've tried to make up my mind on how path-integrals and measure theory are related. I've managed to find the following:
For example if the path $\gamma$ is smooth, if $\gamma^\star$ denotes the image of $\gamma$ and if $\lambda$ denotes the 1-dimensional Lebesgue-measure, then the pushforward
$$ m:= \left( \gamma' \lambda\right)^\gamma$$
is a complex measure on $\gamma^\star$ which satisfies
$$\int_\gamma f(z) dz = \int f d m$$
for appropriate $f$.
My question is: Does anyone know of a (good and quotable) reference where the interplay between path-integration and measure-theoretic integration is examined?
Thanks in advance!