Measured expanders

By measured graphs we mean graphs endowed with a measure on the set of vertices. In this context, we explore the relations between the appropriate Cheeger constant and Poincar\'{e} inequalities. We prove that the expected inequalities hold in two cases: when the measure comes from a random walk, or when the measure has a bounded measure ratio. Finally, we prove that our measured (asymptotic) expanders are Tessera's generalised expanders. This concept is motivated by analytic work in [9] and [10].