I just read the nice exposition [Fermionic Path Integral][1] on nLab and began to wonder about some details to which references appear to be lacking. Suppose we live on Euclidean space as in the Osterwalder-Schrader approach to QFT:
- Is there a deeper analogy between fermionic and bosonic integration? For bosons, we should consider e.g the space $\mathcal{S}'$ of Schwartz distributions as "path space". What is the corresponding - presumably non-commutative - fermionic space $\mathcal{F}'$?
- If such an analogy exists, what is the corresponding one between probability measures $\mu$ on $\mathcal{S}'$ and Berezin integrals $\nu$ on $\mathcal{F}'$?
- Is there an analogy to the spaces physicists would __like__ to work on? i.e The space $\mathcal{S}$ of Schwartz functions would be nice and obliterate the need for regularisation/renormalisation, but unfortunately we have to work on $\mathcal{S}'$ instead. If such an analogy exists, what is $\mathcal{F}$?
- Finally, what features should an interaction $S^{\mathrm{int}}$ have in order to make
\begin{equation}
S^{\mathrm{eff}} \left( \phi \right) = - \ln \int_{\mathcal{F}'} \exp \left[ - S^{\mathrm{int}} \left( \phi, \psi \right) \right] \mathrm{d} \nu_{\mathrm{Berezin}} \left( \psi \right)
\end{equation}
well-defined? i.e How to integrate out a fermion?

__EDIT:__
For the analogies, I was expecting something along the lines:
- For bosons, we encode $\mathbb{R}^4$ by a commutative ring of test functions on $\mathbb{R}^4$, e.g $\mathcal{S}$
- For fermions, I would expect something similar to give a non-commutative ring $\mathcal{F}$
- For bosons, consider the cylindrical measure of a free theory on $\mathcal{S}$ which extends to a measure on $\mathcal{S}'$
- For fermions, consider some weird (in some sense positive) linear functional on $C_b( \mathcal{F} )$ that is somehow compatible with the non-commutativity of $\mathcal{F}$, expose a failure of regularity and show that the failure disappears upon prolonging to $C_b( \mathcal{F}' )$ where $\mathcal{F}'$ is something even more untangible than $\mathcal{F}$ - but has a concrete description

  [1]: https://ncatlab.org/nlab/show/fermionic+path+integral