The following is very important in my research: suppose that $U_t, V_t$ are independent real stochastic processes defined with respect to the same filtration and probability space
Suppose that for $r,s >0$ I need to estimate $\mathbb{E}\left \{ \left( \mathbb{E}[g(U_r,V_r)]-h(U_r) \right)\left( \mathbb{E}[g(U_s,V_s)]-h(U_s) \right) \right\}$ for some regular functions $h,g$.
Suppose that for fixed $u$ the following inequality holds $$\mathbb{E}[ g(u,V_r)]-h(u) \leq C e^{-r} $$ for some constant $C$ independent of $u$ and for every $r>0$.
Can I proceed in the following way?
Consider $(U_\tau= \overline U_\tau, \tau \leq r \vee s)$ to be the sigma algebra generated by $U_\tau$ and compute for some deterministic continuous function $\overline U_\tau$ \begin{align} \mathbb{E} & \left \{ \left( \mathbb{E}[g(U_r,V_r)]-h(U_r) \right)\left( \mathbb{E}[g(U_s,V_s)]-h(U_s) \right) \right\} \\ & = \mathbb{E}\left \{ \mathbb{E} \left \{ \left( \mathbb{E}[g(U_r,V_r)]-h(\overline U_r) \right)\left( \mathbb{E}[g(U_s,V_s)]-h(\overline U_s) \right) |(U_\tau= \overline U_\tau, \tau \leq r \vee s)\right\} \right \}\\ & = \mathbb{E}\left \{ \mathbb{E} \left \{ \left( \mathbb{E}[ \mathbb{E}[g(\overline U_r,V_r) |(U_\tau= \overline U_\tau, \tau \leq r \vee s)]]-h(\overline U_r) \right) \\ \times \left( \mathbb{E}[ \mathbb{E}[g(\overline U_s,V_s) |(U_\tau= \overline U_\tau, \tau \leq r \vee s)]]-h(\overline U_s) \right) |(U_\tau= \overline U_\tau, \tau \leq r \vee s)\right\} \right \}\\ & = \mathbb{E}\left \{ \mathbb{E} \left \{ \left( \mathbb{E}[ \mathbb{E}[g(\overline U_r,V_r) |(U_\tau= \overline U_\tau, \tau \leq r \vee s)]-h(\overline U_r)] \right) \\ \times \left( \mathbb{E}[ \mathbb{E}[g(\overline U_s,V_s) |(U_\tau= \overline U_\tau, \tau \leq r \vee s)]-h(\overline U_s)] \right) |(U_\tau= \overline U_\tau, \tau \leq r \vee s)\right\} \right \} \end{align} and now I want to apply the inequality (since u has been fixed by conditioning) \begin{align} \mathbb{E} & \left \{ \left( \mathbb{E}[g(U_r,V_r)]-h(U_r) \right)\left( \mathbb{E}[g(U_s,V_s)]-h(U_s) \right) \right\} \\ & \leq C^2 e^{-r} e^{-s} \end{align}
In particular is it correct to double condition like that to the same $\overline U$ in both the interior expectation and the external one so that I can use the inequality?