We have the following definition for a conditional expectation:
Let $X: \Omega \rightarrow \mathbb{R}$ be a randomn variable on ($\Omega, \Sigma, \mathbb{P})$. Let $F \subset \Sigma$ be a sub $\sigma-$ algebra, then $Z$ is called a conditional expectation of $X $ under $F$, if $Z$ is $F$ measurable and $\mathbb{E}(X \chi_C) = \mathbb{E}(Z \chi_C)$ for all $C \in F$. What I don't understand here is: What is wrong with $Z=X$? In this case $X$ is $F$ measurable as $X$ is $\Sigma$-measurable and we certainly have $\mathbb{E}(X \chi_C) = \mathbb{E}(X \chi_C)$, so somehow I don't trust this definition. Did I copy anything wrong from the blackboard?