For some application, I'm wondering whether there exists some stochastic process $X:[0,\infty)\times[0,1]\to\mathbb R$ such that for every measurable partition of $[0,1]$ into $\bigcup_{i=1}^nA_i$ -- i.e. $A_i\in\mathcal B[0,1]$, $i\in[n]$, are disjoint Borel sets whose union is the unit interval -- it holds that the processes $Y_i(\cdot)=\int_{A_i}X(\cdot,x)\ \mathrm dx$ constitute independent Brownian motions.
I know of cylindrical Brownian motions, but when we use this construction, we either have a correlation structure on $[0,1]$; or we treat every $x\in[0,1]$ as a separate coordinate, in which case the underlying space would not be separable anymore.
Of course, by Kolmogorov's extension theorem, we can define a process $X(t,x)$ such that $X(\cdot,x)$ is a Brownian motion independent of $X(\cdot,y)$, $y\neq x$, but with such a construction there is no way to assure that the integrals $Y_i(\cdot)$ are well-defined.
I'm looking for either a reference or some direct answer.