# Exponential decay and integration

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.