I found the following in He, Wang, Yan's "Semimartingales":
It is well known that for any non-negative random variable $X$ one can define conditional expectation by $\mathbb E\left[X\mid\mathcal G\right]=\lim_\infty \mathbb E\left[X\wedge n\mid\mathcal G\right]$. However, even if $X$ takes finite values, $\mathbb E\left[X\mid\mathcal G\right]$ may be $+\infty$ on a set with positive probability.
I wonder why it is true since conditional expectation is an "average", if $X$ take finite values,so is the average. Can anyone give my a counterexample?