I am reading Topics in Metric Number Theory, and I want to how to interpret some notation correctly.

On page 44, a metric outer measure is defined via $\delta$-covers as follows. Let $\zeta$ be a premeasure mapping a collection $\mathscr{C}$ of subsets of $\mathbb{R}^d$ to $[0,\infty]$ (satisfying $\zeta(\emptyset)=0$). Then the function $\zeta_*$ defined on the power set of $\mathbb{R}^d$ by

$$ \zeta_*(E) = \lim_{\delta\downarrow0}\uparrow \zeta_{\delta}(E), \text{with} \ \zeta_{\delta}(E)=\inf \sum_{n=1}^{\infty}\zeta(C_n) $$ is an outer measure, where the infimum is taken over all collections $(C_n)_n$ of sets in $\mathscr{C}$ that cover $E$ such that the diameter of $C_n$ is at most $\delta$.

In the above discussion, I understand that $$\lim_{\delta\downarrow0}$$ refers to the limit as $\delta$ goes to zero from above, but I don't know what the "up arrow" in the limit means. I would guess that it means that as $\delta$ decreases to zero, the values of $\zeta_{\delta}(E)$ are increasing, but considering the context this interpretation is counterintuitive.

What is the strict defition of the up arrow $\uparrow$ as used above, and why is it used here?

  • $\begingroup$ That is odd, yes, since $\zeta_\delta$ actually decreases as $\delta$ decreases... $\endgroup$
    – Ian
    Jul 1 '17 at 0:52
  • $\begingroup$ @Ian, that is what I suspected. The chapter uses the notation in a few other contexts that are similar to this one, so perhaps it has some other interpretation other than 'increasing'? $\endgroup$
    – M_B
    Jul 1 '17 at 0:59

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