A note on tameness of families having bounded variation

Abstract

Abstract. We show that for arbitrary linearly ordered set (X, ≤) any bounded family of (not necessarily, continuous) real valued functions on X with bounded total variation does not contain independent sequences. We obtain generalized Helly's sequential compactness type theorems. One of the theorems asserts that for every compact metric space (Y, d) the compact space BVr (X, Y ) of all functions X → Y with variation ≤ r is sequentially compact in the pointwise topology. Another Helly type theorem shows that the compact space M + (X, Y ) of all order preserving maps X → Y is sequentially compact where Y is a compact metrizable partially ordered space in the sense of Nachbin.

Median pretrees and functions of bounded variation

“…Definition 3.1. [20] Let (X, ≤) be a linearly ordered set. We say that a bounded function f : (X, ≤) → R has variation Υ ≤ (f ) not greater than r if…”

“…The following was proved in [20] using the particular case of order-preserving maps and Jordan type decomposition for functions with BV. Theorem 3.2.…”

“…Theorem 3.2. [20] Let (K, ≤) be a compact linearly ordered topological space (with its interval topology). Every function f : K → R with bounded variation is fragmented.…”

“…Weaker versions of these theorems for linearly ordered spaces and BV functions proved in [20], for median pretrees and monotone functions in [13].…”

“…Our aim is to introduce functions of bounded variation on median algebras and pretrees. This was motivated by recent papers [20,13] and especially by [13,Remark 4.11], where we deal with some applications of median pretrees in topological dynamics.…”