Expectation of stationary and ergodic process

Question

Suppose $\{y_t\}$ is stationary ergodic process and $\mathbb{E}[y_t|y_{t-j}, y_{t-j-1}, ...] \rightarrow _{m.s.} 0$ as $j \rightarrow \infty$. Is $\mathbb{E}[y_t] = 0?$

I tried to find a counterexample for this statement, but failed. Thus it seems that this is correct, but I can not prove it.

Answer 1

Stationarity implies that $Ey_t$ is independent of $t$. Since $E(y_t|t_{t-j},...) \to 0$ in the mean we get $Ey_t \to 0$. Hence $Ey_t=0$ for all $t$.