To obtain the time average of an unsteady term like $\frac{\partial u_{i}}{\partial t}$ by definition we perform the following:
\begin{align} \overline{\frac{\partial u_{i}}{\partial t}} &= \frac{1}{T}\int_{t}^{t+T} \frac{\partial }{\partial t}(U_i + {u}'_i)\, dt \\ & = \frac{U_i(x,t + T) - U_i(x,t)}{T} + \frac{u'_i(x,t + T) - u'_i(x,t)}{T} \end{align}
where $U_i$ is the mean value of velocity in $x$-direction and $u'_i$ is the fluctuating part.
My question is that why this term $\frac{u'_i(x,t + T) - u'_i(x,t)}{T}$ equals zero so that $$ \overline{\frac{\partial u_{i}}{\partial t}} = \frac{U_i(x,t + T) - U_i(x,t)}{T} = \frac{\partial U_{i}}{\partial t}$$
Somehow the reason is because $T$ effectively approaches $\infty$ on the time scale of the turbulent fluctuations so that it equals zero, but why isn't that the case for the first term?