Is ergodic in mean a property defined only for WSS stochastic processes? I understand the definition of a random process $X(t)$ being ergodic in mean (first-order ergodic) is that the expectation of the sample mean $&lt;u_X&gt;_T=\frac{1}{T} \int_{\frac{-T}{2}}^{\frac{T}{2}} X_w(t)dt $ should converge to the ensemble mean as $T \rightarrow \infty$. Can we define ergodic in mean property for processes which do not have constant mean.

I know that the property of being ergodic in ACF, second-order ergodic, requires WSS assumption.

My question is motivated by the fact that the requirement of WSS is not given in the wiki article describing ergodic process but used in the example illustrating the same.
Time average is defined over a time interval, T which will always result in a constant value. Keeping this constraint in mind, ensemble mean is always forced to be constant for satisfying the condition of ergodicity.