This isn't always possible, but sometimes you can define what's called an improper prior p(µ), such that even if ∫p(µ) is not finite, the posterior distribution

This isn't always possible, but sometimes you can define what's called an improper prior p(µ), such that even if ∫p(µ) is not finite, the posterior distribution p(µ|x) is. A common example is when p(x|µ,v) is a Gaussian dist. with a prior on the mean set to p(µ)=1.