Can the mean of $\operatorname{sinc}(t)=\frac{\sin(t)}{t}$ be calculated over the entire $t$ domain, i.e., $$\int_{a}^{b}\frac{\operatorname{sinc}(t)}{b-a}dt,\text{ where } a=-\infty, \ b=\infty?$$ Can the calculation be approximated somehow using limits? I have a feeling the mean exists by just looking at a graph the function. The function itself isn't defined at $t=0$, therefore the definition $$\operatorname{sinc}(t)=\begin{cases}1,&\quad t=0,\\\frac{\sin(t)}{t},&\quad \mathrm{otherwise},\end{cases}$$ is used.
Can the calculation be split into two or three sections?
The Dirichlet integral is
$$\int_0^{+\infty} \frac{\sin(x)}{x}dx=\frac{\pi}{2}$$
$\implies$
$$\lim_{(a,b)\to(-\infty,+\infty)}\int_a^b\frac{\sin(x)}{x}dx=\pi$$
$\implies$
$$\lim_{(a,b)\to (-\infty,+\infty)}\frac{1}{b-a}\int_a^b \frac{ \sin(x)}{x}dx=0.$$