Two questions on conditional exectation

Question

Consider the random variable $X$ on $(\Omega,\mathcal{F},P)$ and let $\mathcal{G}\subset\mathcal{F}$.

Q1. What is the difference between $E[X]=E[X|\{\emptyset,\Omega\}]$ and $E[X|\sigma(X)]$?

Q2. Which of $E[X|\mathcal{F}]$ and $E[X|\mathcal{G}]$ is "more random"?

Thanks...

Answer 1

Q1. $E[X|\{\emptyset,\Omega\}]=E[X]; \ E[X|\sigma(X)]=XE[1|\sigma(X)]=X$

Q2. $\mathcal{G}\subset\mathcal{F}$. So, $\mathcal{G}$ is coarser "information" and the first conditional expectation is "more random" in this sense. For example, take $\mathcal{G}=\{\emptyset,\Omega\}$.