A 0-1 law of Brownian motion hitting time

Question

Define the first hitting time $\tau^x_A:=\inf\{t>0, B^x\in A\}$, B is a standard Brownian motion, my question is: if $P(\tau_A^x<\infty)>0$, then $P(\tau_A^x<\infty)=1$? For 1, 2 dimensional Brownian motion, it is true. But what about dimension $\geq 3$?