arXiv:2608.14272v1

Banach's principle in vector lattices

Alexander Dobrick

math.FAmath.DS46B4246A4047B6547A3537A30

Paper metadata

Original submission: August 14, 2026
Claimed MSC: 46B42, 46A40, 47B65, 47A35, 37A30
arXiv categories: math.FA, math.DS
Paper license: CC BY 4.0 ↗

Abstract

We develop an abstract Banach principle in vector lattices and apply it to obtain lattice-theoretic versions of the individual and maximal ergodic theorems, without recourse to any measure representation. Considering a sequence of bounded operators with values in a Dedekind $σ$ -complete vector lattice endowed with a locally solid topology satisfying the $σ$ -Lebesgue property, we prove that the set of points at which the sequence is order convergent is a closed subspace and coincides with the whole space whenever convergence holds on a dense subset. Investigating positive, power-bounded, mean ergodic operators on order continuous Banach lattices, we construct a topology on the universal completion induced by a strictly positive order continuous functional and prove that the Cesàro means converge in order in the universal completion and, in particular, uo-converge in the original lattice. Moreover, we introduce the notion of a superinvariant pair and derive a lattice-theoretic Hopf inequality together with a weak type estimate via band projections, which yields an abstract maximal ergodic theorem. A spectral-theoretic version of the theorem follows from classical Perron--Frobenius theory. Finally, we specialise the abstract framework to the model space $L^0(Ω)$ and revisit the classical Banach principle, the Hopf--Dunford--Schwartz theorem and Doob's martingale convergence theorem.

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Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The abstract Banach principle, the individual and maximal ergodic theorems in vector lattices, the Perron–Frobenius corollary, and the stated classical specializations are correct. One overbroad auxiliary assertion about $L^0$ outside the section's $\sigma$ -finite setting requires a local hypothesis correction but does not affect any main result.

Theorem 2.5Correct

The lattice-valued Banach principle holds

Pages 6–7 · Theorem 2.5 · arXiv:2608.14272v1

Pointwise order boundedness makes the maximal operator well-defined. Baire category and the metrizable locally solid topology make that operator continuous at zero. The tail oscillation criterion then shows that the order-convergence set is a closed linear subspace, so convergence on a dense subset extends to the whole Banach space.

Theorems 3.7 and 4.6Correct

The individual and maximal ergodic theorems are correct

Pages 11–16 · Theorems 3.7 and 4.6 · arXiv:2608.14272v1

A strictly positive functional induces a locally solid gauge topology with the $\sigma$ -Lebesgue property on the universal completion. The dense subspace $\operatorname{fix}T\oplus(I-T)E_e$ has explicit order convergence, and the abstract Banach principle extends it once the Cesàro orbit is order bounded. The superinvariant-pair argument proves the needed weak-type band estimate; lateral completeness assembles the disjoint level bands into a single order bound. Norm mean ergodicity then identifies the order limit with the mean-ergodic projection.

Corollaries 4.14, 5.7, and Proposition 5.10Correct

The spectral and classical applications follow

Pages 17 and 20–22 · Corollaries 4.14, 5.7, and Proposition 5.10 · arXiv:2608.14272v1

Irreducibility turns nonzero invariant positive vectors and functionals into a weak unit and a strictly positive functional. For a Dunford–Schwartz operator on a finite measure space, the constant one and integration form a superinvariant pair, and unbounded-order convergence in $L^1$ becomes almost-everywhere convergence in $L^0$ . Doob's maximal inequality and the dense union of the filtration subspaces satisfy the abstract Banach principle and give the stated martingale convergence.

02Proofs4 reported findingsCorrect

The central proofs are correct and complete. The oscillation, Baire-category, gauge-topology, band-projection, and universal-completion arguments close their stated obligations. Proposition 5.1 should retain a localizability or $\sigma$ -finiteness hypothesis; this is a local scope correction and every downstream application already lies in the $\sigma$ -finite setting.

Theorem 2.5Correct and complete

The Baire-category and oscillation argument is complete

Pages 5–7 · Lemmas 2.1, 2.4 and Theorem 2.5 · arXiv:2608.14272v1

Dedekind $\sigma$ -completeness supplies every countable tail supremum and infimum. The $\sigma$ -Lebesgue property makes the finite maxima converge in the locally solid topology, and a generating Riesz pseudonorm permits the uniform-boundedness argument. Continuity of the maximal operator sends the tail oscillation of a norm limit to zero, which is exactly the order-Cauchy criterion proved in Lemma 2.1.

Theorems 3.7 and 4.6Correct and complete

The universal-completion and maximal-band arguments are complete

Pages 9–16 · Sections 3–4 · arXiv:2608.14272v1

The ideal and regular-sublattice properties transfer the required bounded pieces between $E$ and $E^u$ . The gauge topology is Hausdorff because the chosen functional is strictly positive and the weak unit remains a weak unit in $E^u$ . In the maximal theorem, the finite Hopf inequality passes to increasing bands by $\sigma$ -order continuity; the decreasing level bands have zero intersection, and lateral completeness supplies a dominating element. These steps establish exactly the maximal hypothesis used by Theorem 3.7.

Proposition 5.1Minor formal correction

Universal completeness of $L^0$ requires localizability

Page 18 · Proposition 5.1 · arXiv:2608.14272v1

Read the proposition with the section's standing $\sigma$ -finite hypothesis, or replace 'measure space' by 'localizable measure space.' For a general measure space, $L^0(\mu)$ need not be Dedekind complete; universal completeness is equivalent to localizability of the measure algebra. The correction is local and harmless because Propositions 5.2–5.4 and every later application explicitly use $\sigma$ -finite or probability spaces, which are localizable.

Kusraev–Tasoev, Theorem 3.10 ↗

Corollary 5.7 and Proposition 5.10Correct and complete

The cited classical inputs are applied within their hypotheses

Pages 20–22 · Section 5 · arXiv:2608.14272v1

The finite-measure Dunford–Schwartz hypotheses provide both $L^1$ and $L^\infty$ contractivity and hence the stated superinvariant pair. The martingale application uses Doob's maximal inequality only as an external maximal estimate and correctly limits the abstract principle to its closure step. The filtration-generating assumption makes the union of the adapted $L^1$ spaces norm dense.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.14272v1
Authors listed: Alexander Dobrick
Audit date: August 18, 2026

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Banach's principle in vector lattices

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The lattice-valued Banach principle holds

The individual and maximal ergodic theorems are correct

The spectral and classical applications follow

The Baire-category and oscillation argument is complete

The universal-completion and maximal-band arguments are complete

Universal completeness of L0L^0L0 requires localizability

The cited classical inputs are applied within their hypotheses

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Universal completeness of $L^0$ requires localizability