Problem:
I am self-studying measure theory, Lebesgue integration, and related topics. I have come across this interesting example, which I "know" to be a mixture distribution, but because I have little experience in proofs, I am not sure how to show the result rigorously.
Define:
$F(x)=\begin{cases} 2-3^{-x}-2^{\left\lfloor{x}\right\rfloor} & \text{ if } x\ge 0 \\ 0 & \text{ if } x< 0 \end{cases}$
Show that $F(x)$ is a mixture of discrete and absolutely continuous distributions.
Then, evaluate:
$\int_{\mathbb{R}}e^{-x}F(dx)$ and $\int_{\mathbb{R}}xF(dx)$
Any help is always appreciated. Thanks!