I am trying to understand the notion of pre-Brownian motion using the Book "Brownian Motion, Martingales, and Stochastic Calculus" by J. Le Gall.
He first define the Gaussian white noise to be some isometry $G$ from $L^2(E,\mathcal{E},\mu)$ to a centered Gaussian space, and also in page 12 he states that $G(1_A)$ is a Gaussian R.V. with $\mathcal{N}(0,\mu(A))$, for any Gaussian white noise $G$. Then in page 19, the pre-Brownian motion is defined as a random process $B_t$ such that $B_t=G(1_{[0,t]})$ for some $G$. But for any Gaussian white noise $G$, $G(1_{[0,t]})$ is always $\mathcal{N}(0,\mu(A))=\mathcal{N}(0,t)$, means that $B_t=\mathcal{N}(0,t)$ is uniquely defined.
I don't know if I missed anything.