= Solution
For a partition of $[a,b]$, every <Riemann sum> $S_n=\sum_jB_{t_j}\Delta t_j$ is a centered <normal distribution>[Gaussian random variable], with
$$
\operatorname{Var}S_n=\sum_{i,j}\min(t_i,t_j)\Delta t_i\Delta t_j.
$$
Path continuity gives $S_n\to\int_a^bB_sds$ almost surely, and the covariance bound $\mathbb E|B_s-B_t|^2=|s-t|$ also gives convergence in $L^2$. Hence the limit is Gaussian, centered, and passage to the limit in the displayed sums gives variance
$$
\int_a^b\int_a^b\min(s,t)\,ds\,dt.
$$
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