# Existence of the space of uncountable independent Brownian motions

Does the space satisfying following properties exist：

$$\{\{B^u_t\}_{0\le t<\infty}:u \in [0,1]\}$$,where $$\{B^u_t\}_{0\le t<\infty}$$ is a standard Brownian motion started from $$u$$, and they are mutually independent for $$u$$.

I read Ash,Doléans's book about measure theory in 1999, which says that if I want to construct a product space of uncountable dimension, the "factor" space must have some topological properties. However, I didn't find the results by google.