I am trying to understand better the definition of conditional expectation, for that I want to prove the following:
Let $X$ be a random variable in the probability space $(\Omega, \Sigma, \mathbb{P})$, let $\mathcal{F}$ be a sub-$\sigma$-algebra of $\Sigma$ and let $\mathcal{P}$ be a partition of $\Omega$ such that $\mathcal{P}$ are the minimal generators of $\mathcal{F}$, i.e. $\mathcal{F} = \sigma(\mathcal{P})$.
I want to show that: $$\mathbb{E}_\mathbb{P}(X \mid \mathcal{F}) = \mathbb{E}_\mathbb{P}(X \mid \mathcal{P})$$