A note on tameness of families having bounded variation

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 $BV_r(X,Y)$ of all functions $X \to Y$ with variation $\leq r$ is sequentially compact in the pointwise topology. Another Helly type theorem shows that the compact space $M_+(X,Y)$ of all order preserving maps $X \to Y$ is sequentially compact where $Y$ is a compact metrizable partially ordered space in the sense of Nachbin.