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?


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