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Is there a choice of a $\sigma$-algebra which makes the conditional expectation of an random-variable $X \in L^1_{\mathbb{P}}(\mathcal{F},\mathbb{P})$, on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ equal to the expectation itself?

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    $\begingroup$ Yes, $\mathcal F=\{\Omega ,\emptyset\}$. $\endgroup$ – Surb Feb 5 '19 at 17:04
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If $\mathcal F=\{\Omega,\varnothing\}$ then random variables measurable wrt to $\mathcal F$ must be constant.

$\mathbb E[X\mid\mathcal F]$ is by definition measurable wrt to $\mathcal F$ (so must be constant) and must also satisfy the condition $\mathbb EX=\mathbb E[\mathbb E\mid X]]$.

This together leads to the conclusion that $\mathbb E[X\mid\mathcal F](\omega)=\mathbb EX$ for every $\omega\in\Omega$.

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By identifying $\mathbb{R}$ with the set of constants in $L^1_{\mathbb{P}}(\mathcal{F})$, you get $\mathbb{E}[X]$, (or at-least a copy of it after the identification is made).

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  • $\begingroup$ Cool but, can you please provide some more details...since this is just the same point I'm at....thank you though :) $\endgroup$ – user355356 Feb 5 '19 at 17:05

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