Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 34 / 2 / d / Solution

For $\tau =c\sigma$ with $c>0$, the change of variables givesEquivalently,This is the scale-invariant prior as a measure. Because its integral over $(0,\infty )$ diverges, it is an improper prior, not a normalized probability distribution on that range.