|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|
By pseudo-differential analysis? an elliptic operator acting on sections of two vector bundles on a manifold is a Fredholm operator and hence has closed kernel and cokernel of finite dimension. The difference of these two dimensions is the analytical index of the operator.
More generally, for an elliptic complex, its analytical index is the alternating sum
This index does not the depend of the Sobolev space used to get a bounded operator (by elliptic regularity the kernel is made up of smooth sections and the same for the cokernel as it is the kernel of the adjoint). By topological K-theory one can associate to it also a topological index. The Atiyah-Singer index theorem say that this two indexes coincide.