Solution
= Solution
Choose step functions $f_n\to f$ in $L^2[0,1]$. Part (i) shows that $\int B_sf_n(s)ds$ is centered Gaussian. Since the kernel $K(r,s)=\min(r,s)$ is bounded,
$$
\mathbb E\left|\int_0^1B_s(f_n-f)(s)ds\right|^2
=\iint(f_n-f)(r)(f_n-f)(s)K(r,s)drds\to0.
$$
The $L^2$ limit is therefore centered Gaussian, and its variance is
$$
\int_0^1\int_0^1f(r)f(s)\min(r,s)\,dr\,ds.
$$