Consider a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$, where the filtration $\mathbb{F}$ is generated by a family of partitions $\mathscr{P}=(\mathcal{P}_{t})_{t=0,1,\dots,T}$ of $\Omega$ such that $\mathcal{P}_{t}$ contains more information than $\mathcal{P}_{t-1}$, where $\mathcal{P}_{0}={\Omega}$ and $\mathcal{P}_{t}=\mathcal{P}(\xi_{1},\dots,\xi_{t})$ with $(\xi_{n})_{n\geq 1}$ being i.i.d. random variables. Lastly, let $X$ denote some random variable.
Now what I don't understand is the following equality: $$\mathbb{E}[X\mid \mathcal{P}_{t}]=\mathbb{E}[X\mid \xi_{1},\dots,\xi_{t}].$$ I mean why is it not: $$\mathbb{E}[X\mid \mathcal{P}_{t}]=\mathbb{E}[X\mid \mathcal{P}(\xi_{1},\dots,\xi_{t})]??$$ Feel free to ask for more context if needed.