Linearity of Conditional Expectation

Question

I have read in several places that $E((\sum_{i=1}^{n} X_{i})|Y) = \sum_{i=1}^{n} E (X_{i}|Y)$, but I cannot seem to find a proof for it other than a rough sketch for the continuous case (Linearity of conditional expectation (proof for n joint random variables)). How will one show this result?

Answer 1

What you want to show is that the mapping $X\mapsto \mathbb{E}[X|Y]$ is linear. To do this remember that the characteristic property of the conditional expectation is $$\forall A\in\sigma(Y), \mathbb{E}[X\mathbf{1}_A]=\mathbb{E}\left[\mathbb{E}[X|Y]\mathbf{1}_A\right]$$

Now, if $X,X'$ are two random variables and $\lambda,\mu$ are scalars, the random variable $Z=\lambda \mathbb{E}[X|Y]+\mu\mathbb{E}[X'|Y]$ verifies this property due to the linearity of expectations. By unicity of the conditional expectation (up to sets of probability 0) we have $Z=\mathbb{E}[\lambda X + \mu X'|Y]$.

Once you have this you can simply apply well-known linear algebra properties to get the result for $n$ variable (or proceed by induction).