Let $\mathscr A$ be the $\sigma$-algebra of Borel sets on the unit interval $[0,1]$, and $\mathbb P$ the restriction of the Lebesgue measure on $\mathscr A$. Is there a Wiener process aka Brownian motion on the probability space $([0,1],\mathscr A,\mathbb P)$?
This can of course be reformulated as follows: There is a measurable function $f: [0,1] \rightarrow \mathbb R^{\mathbb R_+}$ such that for any uniform $U$, $f(U)$ is a Brownian motion (!).
Do you have a nice argument for the right answer?
Answer is given in a comment. As there is an i.i.d. sequence of uniforms on this space, we get an i.i.d. sequence of suitable normals which we can indeed use to construct Brownian motion on a dyadic grid and then "continuously" extend it, following P. Lévy. Thanks @geetha290krm.