Suppose $(\Omega,\mathcal F,\mathbb{P})$ is a probability space and that $\mathcal G$ is a sub-sigma-algebra of $\mathcal F$. If $X$ is an integrable, non-negative random variable with the same distribution as $\mathbb{E}[X|\mathcal G]$, how does one show that $X=\mathbb{E}[X|\mathcal G]$ a.s.?
Thank you.
(I can do the case where $X$ has finite variance, but the method doesn't seem to extend to this...)
Edit
I've just found a hint for this question (but remain stuck):
Show that $f(X)=f(\mathbb{E}[X|\mathcal{G}])$, almost surely, where $f(x)$ equals $1$ if $x>0$, $0$ if $x=0$ and $-1$ otherwise.