I'm working on a problem involving a path integral of the form:
$$N = \int X(\omega) dP(\omega).$$
Here, $\omega$ represents sawtooth-shaped paths going from time $0$ to $t$, and each tooth of the sawtooth reaches a height that is independent of the rest of the path. The value of $X$ is multiplicative on the values of the teeth in the sawtooth path, i.e.,
$$X(\omega) = \prod_i X(\omega_i),$$
where $\omega_i$ is the height of the $i$-th tooth of the sawtooth path. The measure $P$ is a path measure on the space of sawtooth paths.
Let p be time-average point measure.: $$p(x)=\frac{1}{t}\int_0^{t} \delta(x-\omega(u))du$$
I believe that given the conditions on the paths, I can relate the function $X(\omega)$ to a function $Y(p)$ and the measures $dP(\omega)$ to $g(p)dp$. Specifically, I would like to set
$$X(\omega) = Y(p) \quad \text{and} \quad dP(\omega) = g(p) dp.$$
With these relations, I can rewrite the path integral as
$$N = \int X(\omega) dP(\omega) = \int Y(p) g(p) dp.$$
My question is: Is this an appropriate application of the Radon-Nikodym theorem? To apply the Radon-Nikodym theorem I want the same event space for P and p. I think it would it be appropriate to say the event space is the probability a tooth obtains a particular value along the path. I believe this is kosher because my teeth are independent, but I'm unsure.
Bonus: This probability theory language seems uncommon for the treatment of path integrals. Are there references for this probability theory treatment of paths? Alternatively, is there a good reason this language is not used for path-integrals?
Any help or suggestions would be greatly appreciated!