I am confronted with the following problem:

Let $\mu$ be a probability measure on $\mathbb{R}$. We wish to show that for any $p \in \mathbb{N}$ and $r \in \mathbb{R}$, the integral $$ F(r):= \int_0^{\infty} |u|^p \exp \bigg\{ - \int_0^u \int_{-\infty}^{z+r} \, \mu(d \theta) \, dz \bigg\} \, du < + \infty.$$ Moreover, we wish to show that $F$ is bounded over $r$.

Since $\mu$ is a probability measure, we clearly have $$ \bigg| \int_0^u \int_{-\infty}^{z+r} \, \mu(d \theta) \, dz \bigg| \leq |u|. $$ Therefore, we are done if there exist results concerning exponential decays of the form $$ \int_0^{\infty} |u|^p \exp \big\{ - \alpha(u) \big\} \, du ,$$ where $\alpha: \mathbb{R} \to \mathbb{R} $ has linear growth. However, I really have problems proving it for the general case.


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