- $E_1$ be a normed $\mathbb R$-vector space
- $E_2$ be a separable $\mathbb R$-Banach space
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $\mathcal F\subseteq\mathcal A$ be a $\sigma$-algebra of $\Omega$
- $X$ be an $\mathfrak L(E_1,E_2)$-valued random variable on $(\Omega,\mathcal A,\operatorname P)$
- $Y$ be an $E_1$-valued random variable on $(\Omega,\mathcal A,\operatorname P)$
I want to show, that if $X\in\mathcal L^1(\operatorname P,\mathfrak L(E_1,E_2))$, $XY\in\mathcal L^1(\operatorname P,E_2)$ and $Y$ is $\mathcal F$-measurable, then $$\operatorname E\left[XY\mid\mathcal F\right]=\operatorname E\left[X\mid\mathcal F\right]\:Y\;.\tag 1$$
If $E_1=E_2=\mathbb R$, then $(1)$ is an elementary result and I know how to prove it. How can we prove it in the more general case described here?
Besides the proof of $(1)$, I wonder if all the basic properties (like the "tower property", etc.) of the conditional expectation in the real-valued case generalize to the Banach space case. Unfortunately, I couldn't find any textbook which gives more than a simple existence/uniqueness result of the conditional expectation in Banach spaces. So, I would be happy if someone could give me a good reference.