Suppose $X$ is a random variable in $(\Omega,\mathcal F)$.$\mathbb E\left(|X|\right)<\infty$. $Y=\mathbb E[X|\mathcal F_0]$ ,here $\mathcal F_0\subset\mathcal F$.
Then I want to show $Y$ is integrable.
The textbook finished this proof by seperating $\Omega$ into $\{Y>0\}$ and $\{Y\le0\}$
But I have tried the following:
According to the definition of the conditional expectation : $$\int_{\Omega}Y \mathrm d \mathbb P=\int_{\Omega}X \mathrm d \mathbb P$$
then $\mathbb E(Y)<\infty$ because $\mathbb E(X)\le\mathbb E(|X|)<\infty$
$\mathbb E(Y)=\mathbb E(Y^+)-\mathbb E(Y^-)$,so both $\mathbb E(Y^+)$ and$\mathbb E(Y^-)<\infty$
then $\mathbb E(|Y|)<\infty$
But the professor said my reasoning was wrong,I don't know which step is false.