Even though there were some questions related to this one, I couldn't find a proper answer. I am wondering if the following inequality holds: $$ \sup_{Z \in \mathcal{Q}} E\left[\varphi^{(1)} E\left[ZS|\mathcal{F}^{(1)}\right]\right]= E\left[\varphi^{(1)}\sup_{Z \in \mathcal{Q}} E\left[ZS|\mathcal{F}^{(1)}\right]\right],$$ where $\varphi^{(1)}$ is a $\mathcal{F}^{(1)}$-measurable random variable with $ \mathcal{F}^{(1)}$ a subfiltration of a general filtration $\mathcal{F}$, $\mathcal{Q} := \{Z \in L^{2}(P,\mathbb{F})\mid E^P[Z|\mathcal{F}^{(1)}]=1 \}$ and $S\in L^{2}(P,\mathbb{F})$.
It is clear that one inequality holds: $$ \sup_{Z \in \mathcal{Q}} E\left[\varphi^{(1)} E\left[ZS|\mathcal{F}^{(1)}\right]\right]\leq E\left[\varphi^{(1)}\sup_{Z \in \mathcal{Q}} E\left[ZS|\mathcal{F}^{(1)}\right]\right].$$ However, I feel that the other inequality holds as well. Here is my argument:
By definition of the supremum, there exists a sequence $Z_n \in \mathcal{Q}$ with $E\left[Z_1 S|\mathcal{F}^{(1)}\right] \leq E\left[Z_2 S|\mathcal{F}^{(1)}\right]\leq \cdots $ such that $\lim_n E\left[Z_n S|\mathcal{F}^{(1)}\right]=\sup_{Z \in \mathcal{Q} }E\left[ZS|\mathcal{F}^{(1)}\right]$. Thus, by the monotone convergence theorem, we have that $$ E\left[\varphi^{(1)} \sup_{Z \in \mathcal{Q}} E\left[ZS|\mathcal{F}^{(1)}\right]\right]=\lim_n E\left[\varphi^{(1)} E\left[Z_n S|\mathcal{F}^{(1)}\right]\right]\\ \leq \sup_{Z \in \mathcal{Q}} E\left[\varphi^{(1)} E\left[Z S|\mathcal{F}^{(1)}\right]\right],$$ which proves the other inequality. Am I right, or there is a mistake in my argument?
Thank you very much in advance!
