DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

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The space of finite total variation signed measures

Definition

Fix a measurable space $(X, A)$ . Write $M (X, A) : = {ν : ν is a signed measure on (X, A) and ∣ ν ∣ (X) < + \infty} .$

For $ν \in M (X, A)$ , define its variation norm candidate by $∥ ν ∥ : = ∣ ν ∣ (X) .$

The next theorem proves that $M (X, A)$ is a real vector space under pointwise addition and scalar multiplication and that $∥ \cdot ∥$ is a norm in the sense of A norm on a real vector space, the induced metric, and the dictionary with the metric axioms.

Depends on

Used by

Dependency tree · two levels

6 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

Sheldon Axler, Measure, Integration & Real Analysis, §9A (standard reference, not scraped)