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Carathéodory measurable sets
Definition
Let be an outer measure on (Outer measures). A set is Carathéodory measurable for when for every .
Write for the family of all Carathéodory measurable subsets of . The quantifier over every test set is part of the definition.
Depends on
Used by
- A three-point outer measure has nonmeasurable subsets despite passing the whole-space split Counterexample
- An outer measure on two points need not be regular Counterexample
- Measurable hulls and regular outer measures Definition
- Counting measure is an outer measure for which every subset is measurable Example
- The zero-one outer measure on a two-point set has only the trivial measurable sets Example
- FALSE: every subset is Carathéodory measurable for every outer measure False statement
- Assuming countable choice, every source-algebra set is measurable for the induced outer measure Lemma
- In the Carathéodory identity, the subadditive inequality is automatic Lemma
- Outer measure splits exactly over finite Carathéodory-measurable partitions Lemma
- Under sigma-finiteness, every Carathéodory measurable set differs from a generated measurable hull by a null set Lemma
- Carathéodory measurable sets form an algebra Proposition
- Closed sets are Carathéodory measurable for metric outer measures Proposition
- Every outer-null set is Carathéodory measurable Proposition
- Carathéodory measurability is not Carathéodory's differentiability criterion Remark
- Assuming countable choice, Carathéodory measurability of a finite-outer-measure set is equivalent to source-algebra approximation Theorem
Dependency tree · two levels
3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- G. Folland, Real Analysis, 2nd ed., Section 1.4 (standard reference, not scraped)
- T. Tao, An Introduction to Measure Theory, Definition 1.7.2 (standard reference, not scraped)