DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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The outer set function induced by a premeasure

Definition

Let $μ_{0}$ be a premeasure on an algebra $A_{0}$ of subsets of $X$ . The set function induced by $μ_{0}$ assigns $E \subseteq X$ the infimum of $\sum_{k} μ_{0} (A_{k})$ over all countable algebra covers $E \subseteq ⋃_{k} A_{k}$ . In symbols,

$μ^{*} (E) : = \inf {\sum_{k = 0}^{\infty} μ_{0} (A_{k}) : A_{k} \in A_{0} and E \subseteq ⋃_{k \in N} A_{k}} .$

The family of covering costs is nonempty because $X \in A_{0}$ and $(X, \emptyset, \emptyset, \dots)$ covers every $E$ . It therefore has an infimum in $[0, + \infty]$ by completeness of the extended real line (Every subset of $\overline{R}$ has a least upper bound and a greatest lower bound in $\overline{R}$ , agreeing with the real supremum and infimum on nonempty sets bounded in $R$ ). This definition calls $μ^{*}$ an outer set function until its outer-measure axioms have been proved.

Depends on

Used by

Dependency tree · two levels

12 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., formula 1.12 (standard reference, not scraped)
T. Tao, An Introduction to Measure Theory, Theorem 1.7.8 (standard reference, not scraped)