Solution
= Solution
For fixed $y\ne x$, part (i), translated by $y$, and then $\varepsilon\downarrow0$ 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>,
$$
\mathbb E\lambda_2(B([0,1]))
=\int_{\mathbb R^2}\mathbb P_x(y\in B([0,1]))dy=0.
$$
The nonnegative random area is therefore zero almost surely.