Solution
= Solution
Approximate $h$ in $L^2[0,t]$ by deterministic step functions $h_n$. Each integral $\int h_n\,dB$ is a linear combination of independent Gaussian increments and hence is Gaussian, with mean zero and variance $\int h_n^2$. The <Itô isometry> gives convergence in $L^2$ to $\int h\,dB$, so characteristic functions pass to the limit. Thus
$$
\int_0^th(s)\,dB_s
\sim N\left(0,\int_0^th(s)^2\,ds\right).
$$