Question about asympotic behavior of $\frac{1}{s}\int_0^s u(x,t) dt$ .

Question

I am just reading a paper, in the final theorem, the author wants to prove that $u(x,t)$ converges to some $v(x)$ in the $L^2$ norm as $t$ $\to$ $\infty$. But in the proof, he defines a $w(x,s)=\frac{1}{s}\int_0^s u(x,t) dt$, and he only proves that as $s$ tends to $\infty$, $w(x,s)$ converges to $v(x)$ in the $L^2$ norm. So my question is: Does $w(x,s)$ represent $u(x,t)$ as $s$ $\to$ $\infty$($t$ $\to$ $\infty$)? Thanks!