Let $X,Y \in \mathcal{L}^1(\mathbb{P})$ be two real valued random variables and $\{\mathcal{F}(t)\}_{t \in \mathcal{T}}$ be a Filtration. If $$\mathbb{E}[X \;\mathbb{1}_{F}]= \mathbb{E}[Y\; \mathbb{1}_{F}]$$ for all $F \in \cup_{t \in \mathcal{T}} \mathcal{F}(t)$ can I conclude that $X= \mathbb{E}[Y|\mathcal{G}] $ a.s., where $\mathcal{G}:= \sigma(\mathcal{F}(t);t \in \mathcal{T})$?
Edit: $X$ is $\mathcal{G}-$measurable