Properties of the intensity (or rate) function of a Non-homogeneous Poisson process

Question

A non-homogeneous Poisson Process is parameterized by its intensity (or rate) function $r(t)$ for $t \in [0, \infty)$. Often what is assumed in the literature about the function $r$ is that $$\int_0^{\infty}r(t)dt = \infty.$$ I understand that this assumption ensures that with probability 1 the process has a jump after any time point $t$. My question is, besides this implication, what would be missing without this unbounded integral assumption on $r$. For instance, if $r$ has only finite support over a closed interval $[0, T]$, is there any reason that one should try to avoid using such a function as the rate function of a Poisson process? Thanks!