# Can a proper prior lead to an improper posterior?

It is my understanding that if we have an improper prior, we may either get a proper or a posterior distribution. My question is if there are cases where a proper prior distribution may, when combined with the likelihood function, result in an improper posterior. Are there any trivial cases? Thanks!

From the sketch of proof https://stats.stackexchange.com/a/89157/14346: if the prior is proper and $p(x|\theta)$ none degenerate then the set of observations for which the posterior is improper is a Lesbegue null set. In other words: $$\mbox{proper prior} \Rightarrow \mbox{proper posterior almost everywhere}.$$ So, there exists some observations $x$ (but a null set) leading to an improper $p(\theta|x)$. An example of such $x$ for a given $p(\theta|x)$ is given here https://stats.stackexchange.com/a/89150/14346.