How does one compute $E[(\int_0^T W_s ds)^2]$ where $(W_t)_{t \in [0,T]}$ is standard Brownian motion in $(\Omega, \mathscr F, \mathbb P)$?
Apparently proving
$$\int_0^T W_s ds = \int_0^T (T-s) dW_s \tag{*}$$
might need to assume that $E[(\int_0^T W_s ds)^2] < \infty$
I think I can just use Ito's lemma to show $(*)$ and then use Ito isometry to show that
$$E[(\int_0^T W_s ds)^2] = E[(\int_0^T (T-s) dW_s)^2] = (\int_0^T (T-s)^2 ds) < \infty$$
Is that okay? If not, why?
Regardless, I'm still curious as to how compute $E[(\int_0^T W_s ds)^2]$ besides using the representation if it's possible? I don't think we encountered $\int_0^T W_s ds$ in classes