 Gaussian change-point posterior with proper priors

Take independent proper normal prior distributions for the two segment means and a proper inverse-gamma distribution for the common variance. A uniform prior distribution over $m=1,\dots ,n$ may then retain the empty second segment. The posterior distribution is proper: its likelihood function is bounded above by $(2\pi {)}^{-n/2}{v}^{-n/2}$, whose integral against $\mathrm{IG}(a_0,b_0)$ is finite for $a_0,b_0>0$. At $m=n$ the second mean has its unchanged proper prior distribution as its full conditional distribution. The configuration's posterior probability now has the meaningful interpretation of no change inside the observed sample.