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Let $X$ be a coarse space, we define the following:

  • $D_b(X)$ is the set of all bounded functions $f:X\rightarrow \mathbb{C}$
  • $f\in $$D_b(X)$ is said to vanish at infinity if for each $\varepsilon$>0 there is a bounded set $K\subseteq X$ such that $\left \| f|_{X\setminus K} \right \|<\varepsilon$, i.e. the set \begin{Bmatrix} x\in X:\left | f(x) \right |\geq \varepsilon \end{Bmatrix} is bounded in $X$. The collection of all functions that vanish at infinity is denoted by $D_0(X)$.
  • Let $f \in D_b(X)$, and $E\subseteq X\times X$ is a controlled set (an entourage), the $E$-variation of f is the function $\operatorname{Var}_E(f)$ defined by $$ \operatorname{Var}_E(f)(x )=\sup\left \{ \left | f(x)-f(y) \right |:(x,y) \in E \right \} $$
  • $f \in D_b(X)$ is said to have vanishing variation if $\operatorname{Var}_E(f)\in D_0(X)$, for all controlled sets $E\subseteq X\times X$.The collection of all functions that have vanishing variation is denoted by $D_h(X)$. I need to verify that $D_0(X)\subseteq D_h(X)$. To this end, let $f \in D_0(X)$, $E$ be a controlled set, and $\varepsilon >0$, then there exists a bounded subset $K\subseteq X$ such that $\left \| f|_{X|K} \right \|<\frac{\varepsilon }{2}$. We aim to show that $\operatorname{Var}_E(f)(x)<\varepsilon$ for all $x\in X|L$, for some $L$ bounded subset. Indeed taking $L=K$, if $x\in X|K$ with $(x,y)\in E$, we consider two cases:
  • Case 1: $y\in X|K$, then $\left | f(x)-f(y) \right |<\varepsilon$.
  • Case 2:$y\in K$, in this case I cannot get that $\left | f(x)-f(y) \right |<\varepsilon$. Can we find a bounded subset $L$ of $X$ such that $ \operatorname{Var}_E(f)(x)<\varepsilon$ outside $L$?
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  • $\begingroup$ Difference between $\operatorname{Var}(f)|_{X-K}$ and $\operatorname{Var}(f|_{X-K})$. $\endgroup$
    – Zach Teitler
    Commented Apr 3, 2020 at 0:03
  • $\begingroup$ $\operatorname{Var}_E(f)(x)$ where $x \in X-K$ is the largest distance between $f(x)$ and $f(y)$ where $(x,y)\in E$, while $\operatorname{Var}_E(f)|_{X-K}(x)$ is the largest distance between $f(x)$ and $f(y)$ where $(x,y)\in E \cap (X-K)\times (X-K)$. How would that help? $\endgroup$
    – Hussain Rashed
    Commented Apr 3, 2020 at 1:01
  • $\begingroup$ Try an example: $X = E = \mathbb{R}$, $f(x) = 1/(1+x^2)$, so $\operatorname{Var}(f)(x) = 1 - 1/(1+x^2)$ for all $x$. Now whatever $K$ is, if $x \notin K$, then $\operatorname{Var}(f)|_{\mathbb{R}-K}(x)$ is still $1-1/(1+x^2)$. The problem is that you shouldn't just take all values $y$. You need to restrict to $y \in X-K$ as well. ... Can I ask you where this question came from? $\endgroup$
    – Zach Teitler
    Commented Apr 3, 2020 at 1:22
  • $\begingroup$ I am reading this paper arxiv.org/pdf/1711.06836.pdf, and it stated on page 16 that $D_0(X)$ is a subset of $D_h(X)$, and that is not clear to me. $\endgroup$
    – Hussain Rashed
    Commented Apr 3, 2020 at 1:36
  • $\begingroup$ At this point it may be a question for the authors. $\endgroup$
    – Zach Teitler
    Commented Apr 3, 2020 at 14:37

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