From Wikipedia:
A compound probability distribution is the probability distribution that results from assuming that a random variable $X$ is distributed according to some parametrized distribution $F$ with an unknown parameter $\theta$ that is again distributed according to some other distribution $G$. The resulting distribution $H$ is said to be the distribution that results from compounding $F$ with $G$. [...] Its probability density function is given by:
$p_{H}(x)=\int p_{F}(x|\theta) p_{G}(\theta) d\theta$
Sometimes, it is possible to define $H$ even when $G$ is not a distribution, by using
$p'_{H}(x) = \int p_{F}(x|\theta) f_{G}(\theta) d\theta$,
$p_{H}(x) = \dfrac {p'_{H}(x)} {\int p'_{H}(x) dx}$,
where $f_{G}$ is a function such that $p_{H}$ is a well defined density function.
Does this generalization of the compound probability distribution have a name? What about '$G$'? Even though it's not a probability distribution, it kind of serves the role of a prior.
I just found out about improper priors. I believe $G$ would be called an improper prior.