• $E_1$ 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
  • $\mathcal F,\mathcal G$ be $\sigma$-algebras of $\Omega$ with $\mathcal F\subseteq\mathcal G\subseteq\mathcal A$
  • $X$ 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 $X$ is $\mathcal G$-measurable and $Y$ is independent of $\mathcal G$, then $$\operatorname E\left[XY\mid\mathcal F\right]=\operatorname E\left[X\mid\mathcal F\right]\operatorname E\left[Y\right]\;.\tag 1$$ $(1)$ is easy to prove in the case of real-valued random variables $X,Y$. How can we show it in the given case?

  • $\begingroup$ To be clear, I know how I can prove the statement in the case of real-valued $X,Y$, but it's not clear to me how the necessary properties of the conditional expectation generalize to the case described in the question. $\endgroup$ – 0xbadf00d Oct 6 '16 at 20:06
  • $\begingroup$ I guess the downvoters didn't read the question and didn't understand why I ask for the proof of an elementary result. $\endgroup$ – 0xbadf00d Oct 6 '16 at 21:49

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