Method I revisited: extension, continuity, and applications
Abstract
Carathéodory's construction, also known as Method I, turns any weight on a family of sets into an outer measure. The difficulty is the extension problem: showing that the outer measure retains the prescribed weights. We record elementary properties of Method I, beginning with the fact that countable subadditivity on the covering family is exactly the criterion for faithful extension, and use them to prove standard results with only outer measures, σ-algebras, and measures, without premeasures on algebras. In topological spaces, a continuity principle reduces this criterion to finite subadditivity and finite approximation by open and compact sets. Applications include the Lebesgue integral as a measure, Tonelli's theorem, multidimensional Lebesgue--Stieltjes measures, Riesz representation for vector-valued functionals, Kolmogorov extension, mass distributions on nested partitions, and Frostman's lemma.
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BibTeX
@misc{badger2026method,
title = {{Method I revisited: extension, continuity, and applications}},
author = {Matthew Badger},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29944},
eprint = {2609.29944},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29944},
}
RIS
TY - UNPB
TI - Method I revisited: extension, continuity, and applications
AU - Matthew Badger
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29944
UR - https://arxiv.org/abs/2609.29944
AB - Carathéodory's construction, also known as Method I, turns any weight on a family of sets into an outer measure. The difficulty is the extension problem: showing that the outer measure retains the prescribed weights. We record elementary properties of Method I, beginning with the fact that countable subadditivity on the covering family is exactly the criterion for faithful extension, and use them to prove standard results with only outer measures, σ-algebras, and measures, without premeasures on algebras. In topological spaces, a continuity principle reduces this criterion to finite subadditivity and finite approximation by open and compact sets. Applications include the Lebesgue integral as a measure, Tonelli's theorem, multidimensional Lebesgue--Stieltjes measures, Riesz representation for vector-valued functionals, Kolmogorov extension, mass distributions on nested partitions, and Frostman's lemma.
ER -