Let $S$ be the unit sphere of $C[0,1], \|\cdot\|_{\infty})$, let $(B_{t})_{t}$ the brownian motion.

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

$\mu_{A}:=P\Big(\frac{B}{\| B \|_{\infty}}\in A | \|B\|_{\infty}>t\Big)$ 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 have an idea ?