Exercise 2.10 - Deriving the inverse gamma density

According to the change of variables formula:

We have:

The change of variables formula is a simplified version of the Lebesgue-Rikdon Theorem, which formally addresses the transform between probability measures defined on the same space. In the simplified version, we take the existence of the derivative $\frac{\text{d}x}{\text{d}y}$ for granted. In the general case, such differential is obtained by taking the limit of simple functions that meet the dominance condition. The Lebesgue Theorem was developed to properly define the differential of one probability measure w.r.t. another probability measure. The derived general differential is usually denoted by $\frac{\text{d}{\mu }_1(x)}{\text{d}{\mu }_2(x)}$ where ${\mu }_i$ are probability measures.