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

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The finitely additive integral on ell-infinity

Definition

Let $ν \in b a (P (N))$ . If a finite-range sequence is written using a finite disjoint partition of $N$ as

$s = \sum_{j = 1}^{m} c_{j} 1_{A_{j}},$

define its finitely additive integral provisionally by

$I_{ν}^{0} (s) : = \sum_{j = 1}^{m} c_{j} ν (A_{j}) .$

The well-definedness lemma The finitely additive integral is well-defined and isometric ↗ proves that this does not depend on the displayed representation and extends uniquely and continuously from the dense finite-range subspace to all of $ℓ^{\infty}$ . That extension is denoted $I_{ν} (x) = \int_{N} x d ν$ .

No countably additive integration theorem is being used here.

Depends on

Used by

The finitely additive integral is well-defined and isometric Lemma

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

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