Let (left(K_{t}^{u}, t geq 0 ight)) be a family of (mathbf{F})-predictable processes indexed by (u geq 0)

Let \(\left(K_{t}^{u}, t \geq 0\right)\) be a family of \(\mathbf{F}\)-predictable processes indexed by \(u \geq 0\) (i.e., for any \(u \geq 0, t \rightarrow K_{t}^{u}\) is \(\mathbf{F}\)-predictable).

Prove that \(\mathbb{E}\left(K_{t}^{\tau} \mid \mathcal{F}_{t}\right)=\int_{0}^{\infty} K_{t}^{u} \alpha_{t}^{u} \eta(d u)\).