There's an interesting exercise (page 13, Exercise 11) in Hugo Duminil-Copin's Lectures on the Ising and Potts models on the hypercubic lattice, which states that the following 2 statements are equivalent for strictly positive measures on $\{0,1\}^E$ where $E$ is a finite set of edges
- $\mu'[\omega \vee \omega'] \mu[\omega \wedge \omega'] \ge \mu'[\omega'] \mu[\omega]$ for all $\omega,\omega'$ where $\vee,\wedge$ denotes taking the component-wise max, min.
- $\mu[\omega_e=1|\omega_{E\backslash e}=\psi] \le \mu'[\omega_e=1|\omega_{E\backslash e}=\psi']$ for all edges $e\in E$ and $\psi \le \psi'$ (edge-wise comparison)
It's clear that condition 1 implies condition 2, but I'm confused on how one would show that condition 2 implies condition 1. Any insights would be appreciated!
