# $\sigma$-algebra making Conditional Expectation equal to Expectation

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?

• Yes, $\mathcal F=\{\Omega ,\emptyset\}$.
– Surb
Feb 5, 2019 at 17:04

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 $$\omega\in\Omega$$.
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).