Measures and Integrals in Conditional Set Theory

Jamneshan, Asgar; Kupper, Michael; Streckfuß, Martin

doi:10.1007/s11228-018-0478-3

Cited by 7 publications

(20 citation statements)

References 26 publications

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“…Let us define conditional intersection and conditional complement. We follow the presentation in Jamneshan et al [36,Section 2]. Let N|A, M|B ∈ P .…”

Section: Preliminariesmentioning

confidence: 99%

“…We have seen that the set operations in S recover the conditional set operations [15, p. 567]. In [15, Corollary 2.10], it was proved that the conditional power set has the structure of a complete Boolean algebra, which is a fundamental result for basic constructions in conditional topology [15,Section 3] and conditional measure theory [36]. The transfer principle for ACA 0 implies a weaker statement, namely that the power set P has the structure of a σ -complete Boolean algebra, see [47].…”

Section: Set Theorymentioning

confidence: 99%

“…We obtain the following extension of von Neumann's mean ergodic theorem, see [1] for the result in ACA 0 . 36 The conditional Hilbert space L 2 (F|G) was introduced in Hansen and Richard [31] for purposes of financial modeling. Some applications in stochastic analysis of an orthogonal decomposition result are described in Cerreia-Vioglio et al [7].…”

Section: Hilbert Spacesmentioning

confidence: 99%

“…For the development of measure theory in secondorder arithmetic, we refer to Simic [46], Simpson [47] and Yu [52]. In conditional set theory, basic results in measure theory are established in Jamneshan et al [36]. As measure theory in second-order arithmetic is based on Daniell's functional approach to integration, the conditional version of the Daniell-Stone theorem and Riesz representation theorem in [36,Section 5] can be connected to respective theorems in S .…”

Section: Further Connections and Outlookmentioning

confidence: 99%

“…Cheridito et al [10], Drapeau et al. [15,18], Filipovic et al [19], and Jamneshan et al [36,38] for an account. It is thus tempting to aim for a general transfer principle that allows one to 'import' classical theorems into a conditional setting.…”

Section: Introductionmentioning

confidence: 99%

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A transfer principle for second-order arithmetic, and applications

Carl

Jamneshan

2018

JLA

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In the theory of conditional sets, many classical theorems from areas such as functional analysis, probability theory or measure theory are lifted to a conditional framework, often to be applied in areas such as mathematical economics or optimization. The frequent experience that such theorems can be proved by 'conditionalizations' of the classical proofs suggests that a general transfer principle is in the background, and that formulating and proving such a transfer principle would yield a wealth of useful further conditional versions of classical results, in addition to providing a uniform approach to the results already known. In this paper, we formulate and prove such a transfer principle based on second-order arithmetic, which, by the results of reverse mathematics, suffices for the bulk of classical mathematics, including real analysis, measure theory and countable algebra, and excluding only more remote realms like category theory, set-theoretical topology or uncountable set theory, see e.g. the introduction of Simpson [47]. This transfer principle is then employed to give short and easy proofs of conditional versions of central results in various areas of mathematics, including theorems which have not been proven by hand previously such as Peano existence theorem, Urysohn's lemma and the Markov-Kakutani fixed point theorem. Moreover, we compare the interpretation of certain structures in a conditional model with their meaning in a standard model. 03C90,03F35; 28B20,54C65

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“…Let us define conditional intersection and conditional complement. We follow the presentation in Jamneshan et al [36,Section 2]. Let N|A, M|B ∈ P .…”

Section: Preliminariesmentioning

confidence: 99%

Section: Set Theorymentioning

confidence: 99%

Section: Hilbert Spacesmentioning

confidence: 99%

Section: Further Connections and Outlookmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A transfer principle for second-order arithmetic, and applications

Carl

Jamneshan

2018

JLA

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An uncountable Furstenberg–Zimmer structure theory

Jamneshan

2022

Ergod. Th. Dynam. Sys.

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Furstenberg–Zimmer structure theory refers to the extension of the dichotomy between the compact and weakly mixing parts of a measure-preserving dynamical system and the algebraic and geometric descriptions of such parts to a conditional setting, where such dichotomy is established relative to a factor and conditional analogs of those algebraic and geometric descriptions are sought. Although the unconditional dichotomy and the characterizations are known for arbitrary systems, the relative situation is understood under certain countability and separability hypotheses on the underlying groups and spaces. The aim of this article is to remove these restrictions in the relative situation and establish a Furstenberg–Zimmer structure theory in full generality. As an independent byproduct, we establish a connection between the relative analysis of systems in ergodic theory and the internal logic in certain Boolean topoi.

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Untitled

2024

Comm. Amer. Math. Soc.

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Let Γ be a countable abelian group. An (abstract) Γ-system X -that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of Γ -is said to be a Conze-Lesigne system if it is equal to its second Host-Kra-Ziegler factor Z 2 (X). The main result of this paper is a structural description of such Conze-Lesigne systems for arbitrary countable abelian Γ, namely that they are the inverse limit of translational systems 𝐺 𝑛 /Λ 𝑛 arising from locally compact nilpotent groups 𝐺 𝑛 of nilpotency class 2, quotiented by a lattice Λ 𝑛 . Results of this type were previously known when Γ was finitely generated, or the product of cyclic groups of prime order. In a companion paper, two of us will apply this structure theorem to obtain an inverse theorem for the Gowers 𝑈 3 (𝐺) norm for arbitrary finite abelian groups 𝐺.

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Measures and Integrals in Conditional Set Theory

Cited by 7 publications

References 26 publications

A transfer principle for second-order arithmetic, and applications

A transfer principle for second-order arithmetic, and applications

An uncountable Furstenberg–Zimmer structure theory

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