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For a random vector $X=(X_1, ..., X_n)^\intercal$ the expectation value can be written as $\mathbb{E}[X] = (\mathbb{E}[X_1], ..., \mathbb{E}[X_n])^\intercal$ according to equation 2 in https://en.wikipedia.org/wiki/Multivariate_random_variable

My question is now how this would look like if we have a conditional expectation regarding another random vector $Y=(Y_1, ..., Y_p)^\intercal$?

Would it just be $\mathbb{E}[X\mid Y] = (\mathbb{E}[X_1 \mid Y], ..., \mathbb{E}[X_n \mid Y])^\intercal$?

If so, why?

Maybe someone could give a measure theoretic reasoning in terms of the generated sigma algebra.

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Yes, $\mathbb{E}[X\mid Y]$ would be exactly as you stated. This is because any element of $X$ could be dependent on any combination of elements of $Y$.

I can't really see what the alternative could be, to be honest

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