My thought was to consider $\displaystyle\lim_{N\rightarrow\infty}\sum_{i=1}^N |B_{t_i}-B_{t_{i-1}}|$. I thought that such a limit might diverge to infinity a.s., and I found a paper that showed that by simple contradiction.
Can we smooth out Brownian path or apply a function to is so as to make it have finite length?
E.g. is there an $f$ such that $\displaystyle\lim_{N\rightarrow\infty}\sum_{i=1}^N |f(B_{t_i})-f(B_{t_{i-1}})|<\infty$ a.s.?