Independence of errors Consider the linear model
$y_i = x_i' \beta+u_i$ for $i=1,\ldots,n$
with $E(y_i \mid x_i)=x_i' \beta \iff E(u_i \mid x_i)=0$. Assume that the observations on $(y_i, x_i')$ are independent over $i=1,...,n$
The textbook claims that $E(u_i \mid x_i,\ldots,x_n)=E(u_i \mid x_i)$. Why is this? How does knowing that $(y_i, x_i')$ is independent from $(y_j, x_j')$ tell us that $E(u_i \mid x_j)=0$?

Independence survives functions. I.e., if random variables $X$ and $Y$ are independent, so are $f(X)$ and $g(Y)$. See this thread for a precise discussion.
And $u_i=y_i−x_i'\beta$ is a function of $(y_i,x_i)$.
