|analytic integration||cohomological integration|
|measurable space||Poincaré duality|
|measure||orientation in generalized cohomology|
|volume form||(virtual) fundamental class|
|Riemann/Lebesgue integration of differential forms||push-forward in generalized cohomology/in differential cohomology|
For the infinite-dimensional version see fermionic path integral.
An exposition of the standard lore is here:
A general abstract discussion in terms of D-module theory is in