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Outer measures
Definition
An outer measure on a set is a function that vanishes at the empty set, is monotone, and is countably subadditive. Explicitly:
- ;
- if , then ;
- for every sequence of subsets of , where the sum is the nonnegative extended sum of Series in the nonnegative extended real line.
The domain is the whole power set . No measurability condition is imposed before is defined.
Depends on
Used by
- A property holding outside a set of elementary measure zero is exactly a property holding λ-almost everywhere Corollary
- Every subset of ℝⁿ has a G_δ measurable hull of the same outer measure Corollary
- 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
- Carathéodory measurable sets Definition
- Measurable hulls and regular outer measures Definition
- Metric 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 outer measure is countably additive on the whole power set False statement
- FALSE: every subset is Carathéodory measurable for every outer measure False statement
- A subset of ℝⁿ with open supersets of arbitrarily small excess is Lebesgue measurable Lemma
- For a Lebesgue measurable set and every positive ε there is an open superset whose difference from it has outer measure below ε Lemma
- In the Carathéodory identity, the subadditive inequality is automatic Lemma
- Every outer-null set is Carathéodory measurable Proposition
- Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure Proposition
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included Theorem
- Assuming countable choice, countable covering costs define an outer measure Theorem
- Assuming countable choice, four equivalent descriptions of a Lebesgue measurable subset of ℝⁿ Theorem
- Assuming countable choice, Lebesgue outer measure is an outer measure that restricts to elementary volume Theorem
- Assuming countable choice, the Lebesgue outer measure of an arbitrary subset of ℝⁿ is the infimum of the measures of the open sets containing it Theorem
- Countable disjoint unions of Carathéodory measurable sets are measurable and split every test set Theorem
- Hausdorff measure is an outer measure Theorem
- The mass distribution principle Theorem
- The RMK functional outer content is an outer measure Theorem
Dependency tree · two levels
10 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 1.7.1 (standard reference, not scraped)