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

Question

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?

Answer 1

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 $\omega\in\Omega$.

Answer 2

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).