Let \(w \in \mathcal{H}=L^{2}(X, \mathscr{A}, \mu)\) and show that \(M_{w}^{\perp}:=\left\{u \in L^{2}: \int u w d \mu=0ight\}^{\perp}\) is either \(\{0\}\) or a one-dimensional subspace of \(\mathcal{H}\).