Let $\mu$ be a measure on $\mathbb{R}$. Suppose $s:=\inf \mathrm{supp}(\mu)=\inf\{x\in\mathbb{R} : \mu(x)>0\}$ is finite.
Then, I want to show following equation for $a\in\mathbb{R}$.
\begin{align} \lim_{t\to\infty} \frac{\int e^{-(t+a)x} d\mu(x)}{\int e^{-tx} d\mu(x)} = e^{-as}. \end{align}
I've already showed following equation. \begin{align} \lim_{t\to\infty} \left( \int e^{-tx} d\mu(x)\right)^{1/t} = e^{-s}. \end{align} So, I thought this equation may be applied to first equation, but I cannot do.
Thanks for your help.