I am reading a paper about stochastic control problems in finance and in particular about the problem of expected utility maximization from terminal wealth. I bumped into the following intuitive formulation: \begin{equation} u(x) = \sup_{V_T\in\mathcal{V}(x)} \mathbb{E}[U(V_T)] \end{equation} where $V_T$ stands for the terminal wealth that is constrained to be in the set $\mathcal{V}(x)$, with $x>0$ as the initial capital. $U$ is the utility function. The filtration with which we are dealing is $(\mathcal{H_t})_{t\in[0,T]}$, it satisfies the usual conditions but $\mathcal{H}_0$ is not necessarily trivial.
Then the paper says that the above problem is equivalent to \begin{equation} ess\sup_{V_T\in\mathcal{V}(x)} \mathbb{E}[U(V_T)|\mathcal{H}_0] \end{equation} in the sense that an element of $\mathcal{V}(x)$ attains the supremum of the first one if it attains the ($\omega$-wise) supremum of the second.
Now, I understand the need of conditioning the expectation to $\mathcal{H}_0$ due to the fact that it may not be trivial, but why the essential supremum? Why not conditioning in the first problem as well? And why $\omega$-wise supremum?