Spaces of measurable functions

Niemiec, Piotr

doi:10.2478/s11533-013-0236-6

Cited by 2 publications

(2 citation statements)

References 19 publications

(22 reference statements)

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“…(7) For us the most important property of M(X ) is the following theorem of Bessaga and Pełczyński [7] (see also [8,Theorem VI.7.1]; for generalizations consult [14]). …”

Section: Hilbert Spacementioning

confidence: 99%

Functor of extension in Hilbert cube and Hilbert space

Niemiec

2014

Open Mathematics

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It is shown that if Ω = Q or Ω = 2 , then there exists a functor of extension of maps between Z -sets in Ω to mappings of Ω into itself. This functor transforms homeomorphisms into homeomorphisms, thus giving a functorial setting to a well-known theorem of Anderson [Anderson R.D., On topological infinite deficiency, Michigan Math. J., 1967, 14, 365-383]. It also preserves convergence of sequences of mappings, both pointwise and uniform on compact sets, and supremum distances as well as uniform continuity, Lipschitz property, nonexpansiveness of maps in appropriate metrics. MSC:57N20, 54C20, 18B30 Keywords

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“…(7) For us the most important property of M(X ) is the following theorem of Bessaga and Pełczyński [7] (see also [8,Theorem VI.7.1]; for generalizations consult [14]). …”

Section: Hilbert Spacementioning

confidence: 99%

Functor of extension in Hilbert cube and Hilbert space

Niemiec

2014

Open Mathematics

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“…Moreover, it is known that in general L 0 (A, X) is not homeomorphic to the classical space L 0 ([0, 1], X) of all Lebesgue measurable functions from [0, 1] to X endowed with the topology generated by the convergence in measure, and to our knowledge the question if all the spaces L 0 (A, X) are homeomorphic or not to an Hilbert space is open. Some results in this latter direction have appeared in [11,Theorem 4.9] in the case where µ is a finite nonatomic measure.…”

Section: Definition 1 ([7]mentioning

confidence: 99%

ON THE ADMISSIBILITY OF THE SPACE L₀(<TEX>$\mathcal{A}$</TEX>,X) OF VECTOR-VALUED MEASURABLE FUNCTIONS

Caponetti¹,

Lewicki²,

Trombetta³

et al. 2013

Bulletin of the Korean Mathematical Society

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Abstract. We prove the admissibility of the space L 0 (A, X) of vectorvalued measurable functions determined by real-valued finitely additive set functions defined on algebras of sets.The notion of admissibility introduced by Klee [7] guarantees that a compact mapping into an admissible Hausdorff topological vector space E can be approximated by compact finite dimensional mappings. This notion is very important in degree theory and fixed point theory. It is known that locally convex spaces are admissible (see [10]). There are some classes of nonlocally convex spaces which are admissible. Riedrich in [13] proved the admissibility of the space S(0, 1) of measurable functions and in [12] the admissibility of the space L p (0, 1) for 0 < p < 1. The admissibility of other function spaces has been proved by Mach [6] and Ishii [8]. In [14] it is proved the admissibility of spaces of Besov-Triebel-Lizorkin type. Definition 1 ([7]). Let E be a Haudorff topological vector space. A subset Z of E is said to be admissible if for every compact subset K of Z and for every neighborhood V of zero in E there exists a continuous mapping H : K → Z such that dim(span [H(K)])< +∞ and x − Hx ∈ V for every x ∈ K. If Z = E we say that the space E is admissible.In this paper we deal with spaces of vector-valued measurable functions and, as a major fact, instead of σ-additive measures we consider finitely additive set functions defined on algebras of sets.Let X be a Banach space, Ω a nonempty set, A a subalgebra of the power set P(Ω) of Ω and µ : A → R a finitely additive set function. We prove

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Spaces of measurable functions

Cited by 2 publications

References 19 publications

Functor of extension in Hilbert cube and Hilbert space

Functor of extension in Hilbert cube and Hilbert space

ON THE ADMISSIBILITY OF THE SPACE L₀(<TEX>$\mathcal{A}$</TEX>,X) OF VECTOR-VALUED MEASURABLE FUNCTIONS

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Spaces of measurable functions

Cited by 2 publications

References 19 publications

Functor of extension in Hilbert cube and Hilbert space

Functor of extension in Hilbert cube and Hilbert space

ON THE ADMISSIBILITY OF THE SPACE L0(<TEX>$\mathcal{A}$</TEX>,X) OF VECTOR-VALUED MEASURABLE FUNCTIONS

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ON THE ADMISSIBILITY OF THE SPACE L₀(<TEX>$\mathcal{A}$</TEX>,X) OF VECTOR-VALUED MEASURABLE FUNCTIONS