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I have a question about conditional expectation. Let $Y$ be a random variable over the space $(\Omega,\mathcal{A},\mathbb{P})$ with $\mathbb{E}(|Y|) < \infty$ and let $\mathcal{C}$ be the sub-sigma algebra which contains only the events with probability $0$ or $1$. Is it true that in this case \begin{equation} \mathbb{E}(Y|\mathcal{C}) = \mathbb{E}(Y) ~~ \text{almost surely}? \end{equation} How can I prove it?

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By the definition of conditional expectation, $\mathbb{E}(Y|\mathcal{C})$ is defined as a $\mathcal{C}$-measurable function such that $\int_{D}\mathbb{E}(Y|\mathcal{C})=\int_D Y$ over any $\mathcal{C}$-measurable subset $D$. Since $\mathbb{E}(Y|\mathcal{C})$ is $\mathcal{C}$-measurable, we can say that $\mathbb{E}(Y|\mathcal{C})$ is actually a constant almost everywhere. (just from your definition of $\mathcal{C}$) Since over the total space it is just the total expectation so your claim is valid.

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