Here is a rough sketch of the proof, due to Holley (1974) : starting from any initial configuration on V , one can run a simple Markov chain (the Metropolis algorithm) that uses independent Uniform[0,1] random variables to update the configuration in each step, such that the chain has a unique stationary measure, the given μ. The monotonicity of μ implies that the configuration at each step is a monotone function of independent variables, hence the product measure version of Harris implies that it has positive associations. Therefore, the limiting stationary measure μ also has this property.