Suppose there is a deterministic process $\phi$ in $L^2(R)$. Need to prove that $\int_0^n \phi_u dW_u$ converges in $L^2(P)$ to some $X\in L^2(P)$ as $n\rightarrow\infty$. Also need to show that $E(X|F_t)=\int_0^t \phi_u dW_u$
I think that to prove the first part it is enough to prove that the sequence is Cauchy, and since $L^2(P)$ is complete, it should follow.
Thank you for help.
Indeed it suffices to show that
$$X_n := \int_0^n \phi_u \, dW_u, \qquad n \in \mathbb{N}$$
is a Cauchy sequence in $L^2(\mathbb{P})$. To prove this, use Itô's isometry and the fact that
$$\int_0^{\infty} \phi_u^2 \, du < \infty$$
by assumption. For the second part, note that $(X_t)_{t \geq 0}$ is a martingale, i.e.
$$\int_F X_n \, d\mathbb{P} = \int_F X_t \, d\mathbb{P}$$
for any $F \in \mathcal{F}_t$ and $n \geq t$. Letting $n \to \infty$, the claim follows from the $L^1$-convergence $X_n \to X$.