DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-14

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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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Finitely additive charges and total variation on the power set of N

Definition

A real or complex charge on $P (N)$ is a function $ν : P (N) \to K$ such that $ν (\emptyset) = 0$ and

$ν (A ⊔ B) = ν (A) + ν (B)$

whenever $A$ and $B$ are disjoint. Only finite additivity is required.

For $A \subseteq N$ , its total variation is

$∣ ν ∣ (A) : = \sup {\sum_{j = 1}^{m} ∣ ν (A_{j}) ∣ : A = A_{1} ⊔ \dots ⊔ A_{m}, m \geq 1} .$

Empty cells may be deleted, and $∣ ν ∣ (\emptyset) = 0$ . The vector space $b a (P (N))$ consists of the charges with $∥ ν ∥_{b a} : = ∣ ν ∣ (N) < \infty$ .

Countable additivity is a strictly stronger property and is not part of this definition.

Depends on

The sequence spaces c_0 and ell-infinity

Used by

The finitely additive integral on ell-infinity Definition

Dependency tree · two levels

4 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

Michael Müger, Introduction to Functional Analysis (standard reference, not scraped)