# Conditional expectation of random variable given minimal sigma-algebra generators

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: $$\mathbb{E}_\mathbb{P}(X \mid \mathcal{F}) = \mathbb{E}_\mathbb{P}(X \mid \mathcal{P})$$

• By definition: $\mathbb{E}_P(X\mid \mathcal{P}) = \mathbb{E}_P(X\mid \sigma(\mathcal{P})) = \mathbb{E}_P(X\mid \mathcal{F})$. Actually the way conditional expectation is defined, the only way of interpreting the second argument in $\mathbb{E}_P(X\mid *)$ is as the $\sigma$-algebra generated by it. (i.e. $\sigma(*)$). – Sayantan Apr 11 at 14:51
• Why do you interpret the second argument in $\mathbb{E}_P(X\mid *)$ as the $\sigma$-algebra generated by it? This should be my question – Robert T Apr 11 at 14:54
• If you start from the most general definition of conditional expectation, you define $\mathbb{E}_P(X\mid \mathcal{F})$ only for sub $\sigma$-algebras $\mathcal{F}$ of $\Sigma$. Hence the only interpretation of $*$ in $\mathbb{E}_P(X\mid *)$ has to be $\sigma(*)$. – Sayantan Apr 11 at 15:00
• What is your definition of $\mathbb E_{\mathbb P}(X|\mathcal P)$? – Mike Earnest Apr 11 at 17:27
• I'm using the definition given by Williams - "Probability with Martingales", i.e., $\mathbb E_{\mathbb P}(X|\mathcal F)$ is the random variable $Y$ that is $\mathcal F$-measurable and satisfies that $\int_A Y d\mathbb P = \int_A X d\mathbb P, \forall A \in \mathcal F$. But for $\mathbb E_{\mathbb P}(X|\mathcal P)$ I don't have any definition since$\mathcal P$ is not a $\sigma$-algebra. – Robert T Apr 12 at 7:28