Path integral via discretization
So let me start with what seems to be the point of view of physicists (corrections are highly appreciated since this is what I understood!). Let a quantum system with coordinates $q_a$ and momenta $p_b$ be given satisfying commutation relations $$[q_a,p_b]=i\delta_{ab}.$$
Further suppose the system has a Hamiltonian $H$ which is time-independent. Usually in that setting the path integral is introduced as a means to compute the transition amplitude
$$\langle q',t'|q,t\rangle=\langle q'|e^{-iH(t'-t)}|q\rangle=\int\mathfrak{D}x(t)\exp\left\{iS[x(t)]\right\}$$
This is usually defined by a discretization procedure allied to a Wick rotation to Euclidean time $\tau = it$ to deal with convergence. The right discretization seems to be derived by slicing the time interval, evaluating $\langle q',t'|q,t\rangle$ to first order in $t'-t$, and imposing some ordering convention.
So: a path integral in Physics is defined by the continuum limit of these aforementioned discretizations.
Cylinder set measures
Now there's the mathematicians point of view on which one studies integration over locally convex vector spaces which are infinite dimensional. In that case, if $E$ is such a space we perform two definitions:
Definition: Let $E$ be a locally convex vector space. A cylinder set is defined to be a subset $C\subset E$ of the form $$C=\{x\in E : (\ell_1(x),\dots,\ell_n(x))\in C_0\}$$ where $C_0\subset \mathbb{R}^n$ is a Borel subset and $\ell_k\in E^\ast$ are continuous linear functionals. Equivalently, it is a preimage $C = P^{-1}(C_0)$ of a Borel set under a continuous linear map $P : E\to \mathbb{R}^n$.
Definition: Let $E$ be a locally convex vector space. A cylinder set measure $\nu$ is a nonnegative additive set function defined on the $\sigma$-algebra generated by cylinder sets of $E$ such that for any continuous linear $P : E\to \mathbb{R}^n$ the set function $$\nu\circ P^{-1} : B\mapsto \nu(P^{-1}(B))$$ is countably additive.
Comparison
If we now compare there are a few points to mention:
It seems that cylinder sets capture discretizations. If $x$ is a continuous path, $(\ell_1(x),\dots, \ell_n(x))$ is an $n$-point discretization. To be even more precise in the case of paths we could take $\ell_k(x) = x(t_k)$ for some $t_1,\dots, t_n$ in the interval. In the same way we could take $\ell_k(x) = a_k$ some Fourier coefficient of $x(t)$. I've seem both things done in Physics.
It seems cylinder set measures are in fact a way to define "a measure per discretization". So for each discretization we give a measure - integrate over $n$ points, integrate over $n$ Fourier coefficients, so forth.
Still, the connection doesn't feel complete for me. The issue is that to define a cylinder set measure we must define $\nu$ on the whole algebra generated by cylinder sets.
The Physicist approach seems to do this exactly for a specific collection of cylinder sets. Either for the ones with $\ell_k(x) = x(t_k)$ or for the ones with $\ell(x_k)=a_k$ a Fourier coefficient.
But there are infinitely many other choices of the $\ell_k$ which give rise to many more cylinder sets. And one would need still to define $\nu$ on the $\sigma$-algebra itself.
The question: is there really a relation between cylinder set measures and the Physicist discretization of a path integral? If so, how the relation can be made more precise? If not, why not, considering the similarities? Are my interpretations (1) and (2) right?
For this discussion, please let us consider the Euclidean path integral. So the issue here is not the imaginary exponent.
