Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and \(f, g \in \mathcal{L}_{T}^{2}\). Show that

a) \(\mathbb{E}\left(f \cdot B_{t} \cdot g \cdot B_{t} \mid \mathscr{F}_{s}ight)=\mathbb{E}\left[\int_{s}^{t} f(u, \cdot) g(u, \cdot) d u \mid \mathscr{F}_{s}ight]\) if \(f\), \(g\) vanish on \([0, s]\).

b) \(\mathbb{E}\left[f \cdot B_{t} \mid \mathscr{F}_{s}ight]=0\) if \(f\) vanishes on \([0, s]\).

c) \(f(t, \omega)=0\) for all \(\omega \in A \in \mathscr{F}_{T}\) and all \(t \leqslant T\) implies \(f \bullet B(\omega)=0\) for all \(\omega \in A\).