$x_1(t)$ and $x_2(t)$ are two "mean ergodic" processes, and their means are $\eta_1$ and $\eta_2$ respectively. If we have:
$$x(t)=x_1(t)+cX_2(t)$$
such that $c$ is an independent variable taking the two values $0$ and $1$ with a probability of $\frac{1}{2}$, Is $x(t)$ "mean ergodic"?
Note: I wanted to solve the problem however there a few things that make me confused. First of all, why is $x_2$ written as $X_2$ while the other $x$'s are written in their normal forms? What does it mean? Second, if I understand correctly, the value of $c$ does not depend on $t$. Third, how should I start to prove that $x(t)$ is mean ergodic? I know that "If the mean value of the process can be obtained as an average over time of this single realization, the process $X(t)$ is said to be ergodic with respect to mean value". (Taken from this link, page 5).
It is clear that the average of $X$ converges to $$ \mathrm{E}[X_1(0)] + c\mathrm{E}[X_2(0)], $$ which is a non-constant random variable, so $X$ is not mean-ergodic.