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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.

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