Monotone Spaces and Nearly Lipschitz Maps

A metric space (X, d) is called c-monotone if there is a linear order < on X and c > 0 such that d(x, y) 6 c d(x, z) for all x < y < z in X. A brief account of investigation of monotone spaces including applications is presented. 1 Monotone and σ-Monotone Spaces. In [6] I investigated existence of sets in Euclidean spaces that have large Hausdorff dimension and yet host no continuous finite Borel measure except the trivial one. (Details are provided below.) I constructed such a set within a line and then extracted the following property of the line that makes the construction work. Definition 1.1. Let c > 0. A metric space (X, d) is called c-monotone if there is a linear order < on X such that d(x, y) 6 c d(x, z) for all x < y < z in X, and monotone if there is c > 0 such that (X, d) is c-monotone. Numerous straightforward generalizations of monotonicity are possible, e.g. local or pointwise monotonicity. We shall consider σ-monotone spaces, i.e. spaces that are countable unions of monotone subspaces (with possibly different witnessing constants). This property still yields the desired measuretheoretic properties of the space. Mathematical Reviews subject classification: Primary: 54E35, 54F05; Secondary: 28A80, 26A27