# $\sigma$-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 '19 at 17:04

## 2 Answers

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).

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