Examples of topological field theories are the cohomological field theories as they were initially defined by Witten [1]. Such examples include the Donaldson-Witten theory in 4d or the Gromo-Witten theory in 2d.

These theories contain a bunch of fields and usually one wants to consider the partition function
$$
Z_{CohTQFT} := \int DX ~ e^{-S_{CohTQFT}[X]}
$$
maybe even include some insertions of operators. My question is **why the measure we see specifically in CohTQFTs is claimed to be well-defined or at least to make "some" sense**.

I do not have a reference for this claim but I am sure it is a well-known thing (and will include reference once I find). Some evidence is that in the end one deal with a finite dimensional integral, e.g. Vafa-Witten theory which calculates the generating function of Euler numbers of the moduli space of instantons over some 4-manifold.

For example, in [2] Witten somehow manages to reduce the path integral of the space of connections (for an abelian gauge theory) to a sum over some homology lattice. This gets me very confused since not only he reduces an infinite dimensional integral to a finite dimensional one but also by considering only homology classes it seems we might be loosing a lot of information as well.