I have a feeling that this question is supposed to be easy, but I'm having a hard time with it (probably because I'm not familiar with conditional expectations with respect to $\sigma$-algebras):
Let $\Omega = \{(x, y)\in [0, 1]^2 \mid x\geq y\}$ and $P$ the uniform probability in $\Omega$. Define the $\mathcal{B}(\Omega)$-measurable random variables $X_1, X_2:\Omega\to\mathbb{R}$ with
$$X_1(x, y)=x$$ $$X_2(x, y)=y$$
for every $(x, y)\in \Omega$. Find the explicit formula for the conditional expectation $E[X_2\mid \sigma(X_1)]$ in terms of $X_1$ and $X_2$ (where $\sigma(X_1)=\{X_1^{-1}(A)\mid A \in \mathcal{B}(\mathbb{R})\}$).
I only know the formal definition of conditional expectation, but this problem made me realize that I have no clue how to find the explicit formula in concrete examples.
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