Consider the following linear regression: $$y_t = \beta_0 + \beta_1 x_{t} + u_t$$ Typically, we need to assume (assuming a random sample):
\begin{equation}\label{I}\tag{I} E[u_t]=0, \quad cov(u_t ,x_t)=0 \end{equation} to conclude that the OLS estimator of $\beta= (\beta_0, \beta_1)$ is unbiased. It's also possible to show consistence. We can also show that $E[u_t|x_t]= 0$ implies (\ref{I}).
Now, consider an $AR(1)$ process (Here we don't have a random sample. We can have dependence): $$y_t = \mu + \alpha y_{t-1}+ \epsilon_t$$ Note that making $x_t := y_{t-1}$, we are in the a linear regression context. But I can't understand why classic time series books don't talk about exogeneity assumption $E[u_t|x_t]=$ or even (\ref{I}). They just say that $(u_t)$ is a white noise process.