 Solution

For fixed $y\ne x$, part (i), translated by $y$, and then $\epsilon \downarrow 0$ show that planar Brownian motion has probability zero of ever hitting $y$ before leaving any fixed large disk. Letting the disk radius tend to infinity shows ${\mathbb{P}}_x(y\in B([0,1]))=0$. By Tonelli theorem,The nonnegative random area is therefore zero almost surely.