For a set $S$, we can evaluate its volume $\operatorname{Vol}(S)$ or its measure $\mu(S)$ based on a measure $\mu$.
My question is: In what condition can $\operatorname{Vol}(S)$ and $\mu(S)$ have the same meaning? I'm writing a paper (in computer science) and want to substitute $c \operatorname{Vol}(S)$ for $\int_{r \in S} c \mu(dr)$, where $c$ is a constant and $\mu$ is any one measure. Is the substituting right? or rigorous?
It completely depends on your definition of $\mathrm{Vol}$, since there is no standard definition. Most people would define it as $\int_Sd\mu$, so it's probably fine. If you want to be absolutely rigorous, though, just go ahead and include this definition of $\mathrm{Vol}$ at the beginning of your paper.