I'm trying to figure out the connections between two contructions of Gaussian measure.

Let $(U, \langle\cdot,\cdot\rangle_U)$ be a seprable Hilbert space, and $\mathcal{B}(U)$ be the Borel sigma-algebra.

Definition 2.1, page 10 gave the following definition of Gaussian measure:

A measure $\mu$ on $(U, \mathcal{B}(u), \mu)$ is Gaussian if for all $v\in U$, there are $m_v\in \mathbb{R}$, $\sigma_v\in \mathbb{R}^+$ such that if $\sigma_v>0$,

$$ \mu(u \in U\colon \langle v,u\rangle_U \in A) = \frac{1}{\sqrt{2\pi\sigma_v}}\int_A\exp(-\frac{(s-m_v)^2}{2\sigma_v^2})ds, $$

for all $A\in\mathcal{B}(\mathbb{R})$.

However, I recall from when we construct the Brownian motion. See, page 23 of https://www.springer.com/gp/book/9780387287201

Let $U$ be the space of continuous funcitons $u(t)$ on $[0,1]$, and $D$ be a cylindrical set

$$ D = \{u\in U: (u(t_1), u(t_2), \ldots, u(t_N))\in E\in \mathcal{B}(\mathbb{R}^N)\} $$

Then the measure is defined by

$$ \mu(D) = \int_E \prod_{i=1}^N (\frac{1}{\sqrt{2\pi(t_i-t_{i-1})}} \exp(-\frac{(h_i-h_{i-1})}{2(t_i-t_{i-1})}))dh_1dh_2\cdots dh_N $$

on $0<t_1<t_2<\cdots<t_N<1$.

My questions is: Is it possible to define the infintite Gaussian measure on finite-dimensional distributions similar to we did for Brownian motion? I don't know how to find the connection bewtween the inner product $\langle v,u\rangle_U$ and a finite-dimensional distribution.

constructionthough, and neither of these statements are really constructions. The Wiener measure paragraph states a certain property for a measure to satisfy; it's still difficult work to prove that there exists a (unique) measure on $C([0,1])$ satisfying this property. Likewise, the Hilbert space paragraph just states a property, and in this case one needs some fairly strong conditions on the $m_v, \sigma_v$ in order to be able to prove there exists a measure with that property. $\endgroup$ – Nate Eldredge Mar 1 at 20:344more comments