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 the measure $\mu_r$ defined on $\mathbb{B}(S)$ by

$\mu_r(A):=P\Big(\frac{B}{\| B \|_{\infty}}\in A \Big| \|B\|_{\infty}>r\Big)$ does not converge weakly to any measure as $r \to \infty$. 

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 ?