Poincar\'e inequalities for mutually singular measures
  Using an inverse system of metric graphs as in: J. Cheeger and B. Kleiner, "Inverse limit spaces satisfying a Poincar\'e inequality", we provide a simple example of a metric space $X$ that admits Poincar\'e inequalities for a continuum of mutually singular measures. 
Andrea Schioppa
math.MG math.DG