I was trained in physics and the way that I learned about Delta functionals was as the limit of a sequence of integrals that contain a "Delta sequence" and a nice function, f. That is,
$\lim_{n\to\infty}\int_{-\infty}^{\infty} \delta_n(x'-x) f(x') dx' = f(x)$
The resulting functional was written as
$ \int_{-\infty}^{\infty} \delta(x'-x) f(x') dx' = f(x)$
where the Delta function, $\delta(x'-x)$, is a "generalized function".
I understand it is also possible to define Delta functions and Delta functionals in terms of measures.
I am wondering, is it rigorously acceptable to write the Delta functional as
$\int_{x-a}^{x+b} \delta(x'-x) f(x') dx' = f(x)$
for $a$ and $b$ being some positive numbers? I know when the Delta function is defined as a limit of a sequence of integrals, they always integrate from $-\infty$ to $\infty$. Maybe that is not necessary, or not necessary when the Delta functional is defined in terms of measures? If the answer is that it is OK to specify the Delta functional over a finite interval, then what is the definition/derivation of the Delta function or Delta functional that would allow that?
I suppose that if $f(x) = 0$ for $x \notin [x-a,x+b] $, then it should be OK. How about if that is not the case?
I have seen some people use a finite integration interval and was wondering if they were being too loose with their notation. Curious what the answer is from a mathematician's perspective.