Consider a continuous time point process $\eta(t)$ representing the number of points in the interval $[0, t]$. Let $\eta(\infty)$ be distributed as the total number of children of a particle. Define $\mu(t)=\mathbb{E}[\eta(t)]$ and so $\mu(\infty)=\int_0^{\infty}\mu(dt)$.
The Malthusian parameter, if it exists, is defined as the solution $\alpha$ of
$$ 1=\int_0^{\infty} e^{-\alpha t}\mu(dt). $$
I want to show that if $\mu(\infty)\in(1,\infty)$ then the Malthusian parameter $\alpha$ satisfying the integral must exist but I'm completely unsure how to proceed.
It looks like it's presented as a part of Proposition $2.2$ on page $6$ of this paper but I don't see a proof included. Any hints, sources or other questions on this platform related to this are welcome.
That follows by continuity and intermediate value theorem. We have continuity since by monotonicity
$$|F(a)-F(a+\epsilon)|=\int_{0}^{\infty}e^{-at}(1-e^{-\epsilon t})\mu(dt)\leq \epsilon \int_{0}^{\infty}e^{-at}t\mu(dt)\to 0$$
and similarly for $|F(a)-F(a-\epsilon)|$. And since $F(0)=\mu(\infty)>1$ and $F(+\infty)=0$ we have some $a_{0}$ s.t. $F(a_{0})=1$.