A covering construction of labelled packing measure and content

@misc{liu2026covering,
  title = {{A covering construction of labelled packing measure and content}},
  author = {Peizhi Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26335},
  eprint = {2609.26335},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26335},
}

TY  - UNPB
TI  - A covering construction of labelled packing measure and content
AU  - Peizhi Liu
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26335
UR  - https://arxiv.org/abs/2609.26335
AB  - We introduce labelled packing measure by a covering construction in which each set has an independent positive label bounding its diameter and the radii in its local packing cost. Letting the labels tend to zero defines a metric outer measure M^s. For every s>0 and every metric space (X,d), we prove 2^−sP^s(F)≤M^s(F) ≤ C(s) P^s(F) (F⊂ X), where $ P^s(F)$ is the classical packing measure and C(s) depends only on s. Thus labelled packing measure and classical packing measure have the same null sets and critical exponent. Allowing arbitrary finite positive labels defines labelled packing content. At the common positive Hausdorff and packing dimension, this content and the labelled packing measure agree on cylinders associated with irreducible subshifts of finite type. In particular, equality holds for non-singleton self-similar attractors and components of finite strongly connected graph-directed systems, without separation assumptions.
ER  -