Conditional Expectation over the Trivial Sigma Algebra is Expectation

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space and let $X$ be a Random Variable on $\Omega$. Then

$$\mathbb{E}[X| {\emptyset, \Omega}] = \mathbb{E}[X]$$

Observe that $\mathbb{E}[X]$ is indeed trivially measureable since Constant Functions are Measureable. Furthermore

$$\int\limits_{\Omega} X d \mathbb{P}(\omega) = \mathbb{E}[X] = \int\limits_{\Omega} \mathbb{E}[X]$$

and

$$\int\limits_{\emptyset} X d \mathbb{P}(\omega) = 0 = \int\limits_{\emptyset} \mathbb{E}[X]$$

so $\mathbb{E}[X]$ is the Conditional Expectation of $X$ with respect to ${\emptyset, \Omega}$. $\blacksquare$

• Conditional Expectation
• Expectation
• Trivial Sigma Algebra