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Instability pairs. (English) Zbl 0585.28003

Let \(\mu\) be a capacity on \({\mathbb{R}}^ d\), i.e. a translation-invariant nonnegative nondecreasing Borelian set function. We shorten \(\mu\) B(a,r) to \(\mu\) (r). Let \(\lambda\) be a content on \({\mathbb{R}}^ d\), i.e. a countably quasi-subadditive capacity, and assume \(\lambda\) (r)/\(\mu\) (r)\(\to 0\) as \(r\downarrow 0.\)
Conditions are given on (\(\lambda\),\(\mu)\) which are sufficient to ensure that each of the following hold, for each Borel set E and \(\lambda\) almost all \(a\in {\mathbb{R}}^ d\) (we abbreviate \(E\cap B(a,r)=E(a,r)):\) \[ (A)\quad \limsup_{r\downarrow 0}\frac{\mu E(a,r)}{\mu (r)}>0\quad or\quad \limsup_{r\downarrow 0}\frac{\mu E(a,r)}{\lambda (r)}<+\infty, \]
\[ (B)\quad \lim \inf_{r\downarrow 0}\frac{\mu E(a,r)}{\mu (r)}>0\quad or\quad \lim_{r\downarrow 0}\frac{\mu E(a,r)}{\mu (r)}=0, \]
\[ (C)\quad \limsup_{r\downarrow 0}\frac{\mu E(a,r)}{\lambda (r)}=+\infty \quad or\quad \lim_{r\downarrow 0}\frac{\mu E(a,r)}{\lambda (r)}=0. \] The results are applied to specific examples.

MSC:

28A75 Length, area, volume, other geometric measure theory
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