Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14
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RNP--dentability characterization

Statement

Assume the Axiom of Choice. A Banach space X has the Radon--Nikodym property if and only if every nonempty bounded closed convex subset of X is dentable.

Facts & Assumptions

[A1]

The Axiom of Choice holds (The Axiom of Choice).

[L1]

RNP is the density property for absolutely continuous bounded-variation vector measures (Radon--Nikodym property).

[L2]

Under AC, dentability of every nonempty bounded closed convex set supplies all required vector-measure densities (Dentable average ranges give vector-measure densities).

[L3]

Under AC, any nondentable such set supplies an absolutely continuous bounded-variation Lebesgue vector measure without a Bochner density (Nondentability produces a vector measure without density).

Proof

technique · direct

Given: A Banach space X and AC.

1.1

Prove the dentability-to-RNP implication. If every nonempty bounded closed convex subset of X is dentable, [L2] applies and gives RNP.

givenA1L2
1.2

Prove the RNP-to-dentability implication. Assume X has RNP. If a nonempty bounded closed convex set were nondentable, [L3] would give a finite-measure, absolutely continuous bounded-variation vector measure without a Bochner density, contradicting [L1]. Thus every such set is dentable.

givenA1L1L3
2.1

Combine the implications and close the degenerate case. [A1, step 1.1, step 1.2] Steps 1.1 and 1.2 prove the equivalence. For the zero Banach space the only nonempty bounded closed convex sets are singletons, which are dentable by the zero-functional slice, and its only vector measure has the zero density. All AC use is inherited exactly from [L2] and [L3].

A1step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

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Sources