Figure 10.

Pigou–Dalton transfer relative to q. A distribution $p\in {\mathbb{P}}_{\mathsf{\Omega }}$ is transformed relative to q by first moving some amount of weight $\epsilon \ge 0$ from $P\left(A_n\right)$ to $P\left(A_m\right)$ where $n,m$ are such that $P\left(\omega \right)\le P\left({\omega }^{\prime }\right)$ whenever $\omega \in A_m$ and ${\omega }^{\prime }\in A_n$, with $\epsilon$ small enough such that this relation remains true after the transformation, and then mapping the transformed distribution back to ${\mathbb{P}}_{\mathsf{\Omega }}$ by ${\mathsf{\Lambda }}_q^{-1}$ (see Definition 6).