Let $(\Omega,\Sigma,\mu)$ be a $\sigma$-finite measure space.
Assume we have already shown that a conditional expectation operator $\mathbb{E}(\cdot\vert\mathcal{F}) : L^1(E)\rightarrow L^1(E)$ exists, where $E$ is a Banach space.
Is there anyway to construct the conditional expectation of a function in $L^2(E)$ using this operator?
I know that one can do this the other way around, but I want to know if it is possible in this direction as well. If the measure space was finite this would be trivial. Can one somehow use an exhausting sequence of sets with finite measure to show this? I wasn't able to come up with anything.