# Discretization of a path integral and cylinder set measures

I'm studying Gaussian measures and in particular the Wiener measure, and in the process I've read about cylinder set measures. As in Wikipedia's page they are defined as:

Let $$E$$ be a real separable topological vector space. Let further $$\mathcal{A}(E)$$ be the set of all surjective continuous linear maps onto finite dimensional vector spaces $$T : E\to F_T$$ where $$\dim F_T < \infty$$. A cylinder set measure is a collection of probability measures $$\{\mu_T : T\in \mathcal{A}(E)\}$$ on the finite dimensional vector spaces $$F_T$$ such that whenever there is a surjective map $$\Pi_{ST} : F_S\to F_T$$ we have $${\Pi_{ST}}_\ast\mu_S=\mu_T$$.

So the idea is that we have a collection of probability measures on all the finite dimensional spaces onto which $$E$$ projects.

When Physicists define the euclidean path integral, they usually do so by discretization. In other words, the integral

$$\int \exp\left[ -\frac{1}{2}\int_a^b \dot{x}(t)^2dt\right]\mathfrak{D}x(t)$$

is evaluated by discretization. The idea seems to be that if we take a path $$x(t)$$ and pick $$N$$ points $$t_1\dots t_N\in(a,b)$$ then we can specify the path in this approximation by the points $$x(t_1),\dots, x(t_N)$$. Then one "integrates over $$dx(t_1),\dots, dx(t_N)$$ so that varying the points one varies the path". In the end for each $$N$$ they define $$I_N=C_N\int dx_1\cdots dx_N \exp \left[-\frac{1}{2}\sum_{k=1}^{N+1} \frac{(x_k-x_{k-1})^2}{\epsilon_N^2}\right]$$ where it is understood that $$x_0 = q$$ and $$x_{N+1}=q'$$ are fixed, $$\epsilon_N = (b-a)/(N+1)$$ is the spacing of the discretization and $$C_N$$ is a normalization constant.

Now, I feel that somehow what they are trying to do is to define a cylinder set measure as in the above definition.

It seems like we are taking the space of paths $$E = C^0([a,b],\mathbb{R})$$ and projecting onto $$\mathbb{R}^{N}$$ by $$T_N[x(t)]=(x(t_1),\dots, x(t_N))$$. This is obviously surjective because given $$N$$ points we can linearly interpolate them to get $$x(t)$$ so that $$T[x(t)]$$ gives the $$N$$ points.

Then one would put $$\mu_{N}(U)=C_N\int_U dx_1\cdots dx_N \exp \left[-\frac{1}{2}\sum_{k=1}^{N+1} \frac{(x_k-x_{k-1})^2}{\epsilon_N^2}\right]$$

These are just guesses. One obvious problem is that in the definition of a cylinder set measre we must define a probability measure $$\mu_T$$ on $$F_T$$ for every $$T$$ and the above defines just for a class of $$T$$.

So the question is: is there some connection between the Physicist definition of path integrals via discretization with cylinder set measures? If so, what is the rigorous connection between then?