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I was thinking what is the $\sigma$-algebra generated by the random variable $Z= \mathbb{I}(X+Y=0)$ where $X,Y\sim\operatorname{Bern}(p)$ iid. (Note: $\mathbb{I}$ is the indicator function.)

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It's the smallest $\sigma$-algebra making $Z$ measurable. As the values of $Z$ are $0$ or $1$, $\sigma(Z)=\{\emptyset,\Omega, Z^{-1}(\{0\}),Z^{-1}(\{1\})\}$ and $$Z^{-1}(\{0\})=\{X\neq -Y\}=\{X=1,Y=-1\}\cup\{X=1,Y=-1\}.$$

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Thanks Davide! (btw you have made a typo with the minus signs) Could you also tell me what the conditional expectation of X given Z is? –  adamG Oct 29 '12 at 23:28

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