Dual space of $C^{1,2}_b([0, T]\times O)$

Question

Let $T>0$ and $O\subseteq \mathbb R$ an open set. Let $C^{1,2}_b([0, T]\times O)$ be the space of functions $u$ continuous bounded such that $\partial_t u$, $\partial_x u$ and $\partial_{xx}u$ are continuous and bounded. Endow this space with the norm: $$|||u|||:=\|u\|_\infty + \|\partial_t u\|_\infty + \|\partial_x u\|_\infty + \|\partial_{xx}u\|_\infty.$$ Suppose that $F:C^{1,2}_b([0, T]\times O)\rightarrow \mathbb R$ is linear and continuous, with $F(1)=1$, and such that for $u\in C^{1,2}_b([0, T]\times O) $, $u \geq 0$, we have $F(u)\geq 0$. Can we have a representation of $F$ in terms of a probability measure?