Consider the measurable space $(\Omega,\mathbb{P},\mathcal{F})$ and a filtration $\mathbb{F}=\left\{\mathcal{F}_t\mid\,t\in[0,T]\right\}$. Let $\mathcal{Q}$ be a family of equivalent probability measures respect to $\mathbb{P}$, and $X$ a random variable. Is it possible to define \begin{equation} M_t=ess\sup_{\mathbb{Q}\in\mathcal{Q}}\mathbb{E}^\mathbb{Q}(X\mid \mathcal{F}_t) \end{equation}
and in this case is $M$ a submartingale $\mathbb{Q}\in\mathcal{Q}$? My question arises because the probabilty spaces where the conditional expectations are defined are not the same.