I have a question concerning a claim in J.F. LeGall's book Mouvement brownien, martingales et calcul stochastique.
Let $(\mathcal{F_{t}})$ be the canonical completed filtration on $\Omega$ of a real brownian motion B isssued from $0$. $\mathcal{F_{\infty}}=\sigma(B_{t},t\geq 0)$. By theorem 5.7. there exists for every random variable $Z\in$ $L^{2}(\Omega,\mathcal{F}_{\infty},P)$ a unique process $h\in L^{2}(B)=L^{2}(\Omega\times \mathbb{R}_{+},\mathrm{Prog},dPd\left<M,M\right>_{s})$ such that $$Z=E[Z]+\int_{0}^{\infty}h_{s}dB_{s}.$$
Then it is shown as a consequence that $(\mathcal{F_{t}})_{t\geq 0}$ is right-continuous. For this he takes a r.v. $Z$ which is $\mathcal{F}_{t+}$ measurable and bounded. He claims that one can find $h\in L^{2}(B)$ such that $$Z=E[Z]+\int_{0}^{\infty}h_{s}dB_{s}.$$
WHY does this representation exists, that is why can we apply theorem 5.7?