Let $S$ be the unite sphere of $C[0,1], \left| \left| \right| \right|_{\infty})$, let $(B_{t})_{t}$ the brownian motion and $M:=\left| \left|B \right| \right|_{\infty}\sim \left| N(0,1)\right|$

I would like to show that for any $A\in \mathbb{B}(S)$

$\mu_{A}:=P(\frac{B}{\left|\left| B \right| \right|_{\infty}}\in A |\left| \left|B \right| \right|_{\infty}>t)$ does not converge weakly to any measure as $t \to \infty$.

I tried to evaluate the measure $\mu$ on the subsets of the cylinder generated by the evaluation map but it is still difficult to manipulate.

Does any one has an idea ?