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I'm reading a proposition given without proof in this note.

Proposition 12.4. Let $\mathcal{G}$ be a sub-$\sigma$-field of $\mathcal{F}, X, Y$ be two random variables such that $X$ is independent of $\mathcal{G}$ and $Y$ is $\mathcal{G}$-measurable, and let $\varphi: \mathbb{R}^2 \rightarrow \mathbb{R}$ be a Borel-measurable function such that $\mathbb{E}(|\varphi(X, Y)|)<$ $+\infty$. Then $$ \mathbb{E}(\varphi(X, Y) | \mathcal{G}) = \psi(Y) \quad \text { a.s., } \quad \text { where } \psi(y)=\mathbb{E}(\varphi(X, y)). $$

It follows from the proposition that $\mathbb{E}(\varphi(X, Y) | \mathcal{G}) = \psi(Y) = \mathbb{E}(\varphi(X, Y))$ is a constant.

Question:

Could you elaborate on how to prove this proposition?

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Analyst
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    There should be a proof of this in most texts. See for example chapter 4.1 (maybe 4. 2?) of Durrett PTE 5th edition – Andrew Jan 15 '23 at 01:24
  • @AndrewZhang Example 4.1.7. in Durrett's PTE only contains a special case where $\mathcal G = \sigma(X)$. – Analyst Jan 15 '23 at 02:36
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    So here we are looking at $Y$ is $\mathcal G$ measurable. This holds for any $\sigma(Y)\subset \mathcal G$. There are some properties about conditional expectation comparing conditioning on $\mathcal G$ versus on $\mathcal F$ where one contains the other. I have a hunch that should do the trick. – Andrew Jan 15 '23 at 03:03
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    @AndrewZhang You're right. Could you post your comments as an answer? – Analyst Jan 15 '23 at 03:05
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    What do you mean by $\psi(Y) = \mathbb{E}(\varphi(X, Y))$? They are different: $$\psi(Y(\omega))=\mathbf{E}[\varphi(X,y)]{y=Y(\omega)}=\int{\Omega}\varphi(X(\eta),Y(\omega)),\mathbb{P}(\mathrm{d}\eta)$$ but $$\mathbb{E}[\varphi(X,Y)]=\int_{\Omega}\varphi(X(\eta),Y(\eta)),\mathbb{P}(\mathrm{d}\eta).$$ – Sangchul Lee Jan 15 '23 at 03:52
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    @SangchulLee You are right! I should be more careful with the notation, i.e., $\psi(Y) = \mathbb{E}_{X \sim \mathbb P}(\varphi(X, Y))$. – Analyst Jan 15 '23 at 03:57
  • @AndrewZhang Could you elaborate on your idea of using the Tower property? – Aditya Dhawan Jan 15 '23 at 05:23
  • See for instance Example 4.1.7 and Theorem 4.1.12,13 of Durrett PTE – Andrew Jan 15 '23 at 03:10

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