I'm a physicist working with the Feynman-Kac integral. It is an integral over the Wiener measure. I'm interested in knowing if I can treat is as a Lebesgue integral in the following manner: When using the Lebesgue integral and measure, $\mu_L$, we have that, if $s$ is a measurable simple function and $E$ some set, then $\int_E sd\mu_L=\sum \int_E a_j 1_S d\mu_L=\sum a_j \mu_L(E\cap S)$ with $S$ a collection of disjoint sets.
What I want to know is if this also applies when using a Wiener integral and measure?