Many perturbative QFTs suffer from the lack of a rigorous definition of a "good enough" measure over the space of paths (or fields) $P$.

$$\mathcal{Z} = \int_{{x \in P}} e^{iS(x)} Dx$$

There are many ways to "fix" this. For example, if the gauge group $G$ is big and good enough, one could instead work with the (hopefully finite dimensional) quotient space $P/G$. However this does not work for all cases. I'm wondering if there are approaches that work for suitable subspace $P' \subset P$?

For example, instead of working with all continuous paths, work with piecewise linear paths or paths in Schwartz space (smooth and tamed).

Many perturbative QFTs suffer from the lack of a rigorous definition of a "good enough" measure over the space of paths (or fields) $P$,

$$\mathcal{Z} = \int_{{x \in P}} e^{iS(x)} Dx$$

There are many ways to "fix" this. For example, if the gauge group $G$ is big and good enough, one could instead work with the (hopefully finite dimensional) quotient space $P/G$. However this does not work for all cases. I'm wondering if there are approaches that work for suitable subspace $P' \subset P$?

For example, instead of working with all continuous paths, work with piecewise linear paths or paths in Schwartz space (smooth and tamed).