I have come across some measures, such as the Lioiuville Measure, where the definition is often given in terms of other variables in differential notation. For example: $$d\mu = \prod_i dp_i\,dq_i.$$
I have seen this notation when a measure is defined in terms of another measure. For example, say $\mu$ is a measure, then we may define $d\nu = f d\mu$ for some $\mu$-measurable function $f$ by $$\nu(E) = \int_E f d\mu.$$
However, I have not seen it written solely in terms of variables as opposed to measures. This seems to be common in mathematical physics. In the case of the Liouville measure, we are defining the measure in terms of momenta and position. My intuition tells me that, taking $i = 2$,: $$d\mu = \prod_i d\,p_i d\,q_i \implies \mu = \iint dp_i dq_i.$$
Is this the case? If so, is this integral to be interpreted as a Riemann integral, as we are integrating with respect to a variable as opposed to a measure? This seems unlikely, so I suspect we are instead using the Lebesgue measure. If that is the case, how do we compute such integrals with respect to say momenta?