I'm reading this paper that's talking about the norm of the conditional expectation operator.
He starts by defining $\mathbb{E}[\ \dot \ |\mathcal{F'}]: \ L^2(X,\mathcal{F}) \mapsto L^2(X,\mathcal{F'})$ where $\mathcal{F'}$ is a sub-sigma algebra of $\mathcal{F}$.
Then it says: "if f is also in $L^1(X,\mathcal{F})$, then $sgn(\mathbb{E}[\ f \ |\mathcal{F'}])$ is in $L^2(X,\mathcal{F'})$ and $\langle f - \mathbb{E}[\ f \ |\mathcal{F'}], sgn(\mathbb{E}[\ f \ |\mathcal{F'}]) \rangle=0$."
Why does $sgn(\mathbb{E}[\ f \ |\mathcal{F'}]) \in L^2(X,\mathcal{F'})$ follow from $f \in L^1 \cap L^2$?
(the paper can be found here: http://people.maths.ox.ac.uk/greenbj/papers/conditional-expect.pdf).