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1
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On positive embeddings of C(K) spaces

100%
Plebanek G.
Studia Mathematica
|
2013
|
tom 216
|
nr 2
179-192
EN
We investigate isomorphic embeddings T: C(K) → C(L) between Banach spaces of continuous functions. We show that if such an embedding T is a positive operator then K is the image of L under an upper semicontinuous set-function having finite values. Moreover we show that K has a π-base of sets whose closures are continuous images of compact subspaces of L. Our results imply in particular that if C(K) can be positively embedded into C(L) then some topological properties of L, such as countable tightness or Fréchetness, are inherited by K. We show that some isomorphic embeddings C(K) → C(L) can be, in a sense, reduced to positive embeddings.
2
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Approximating Radon measures on first-countable compact spaces

100%
Plebanek G.
Colloquium Mathematicum
|
2000
|
tom 86
|
nr 1
15-23
EN
The assertion every Radon measure defined on a first-countable compact space is uniformly regular is shown to be relatively consistent. We prove an analogous result on the existence of uniformly distributed sequences in compact spaces of small character. We also present two related examples constructed under CH.
3
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On Pettis integral and Radon measures

100%
Plebanek G.
Fundamenta Mathematicae
|
1998
|
tom 156
|
nr 2
183-195
EN
Assuming the continuum hypothesis, we construct a universally weakly measurable function from [0,1] into a dual of some weakly compactly generated Banach space, which is not Pettis integrable. This (partially) solves a problem posed by Riddle, Saab and Uhl [13]. We prove two results related to Pettis integration in dual Banach spaces. We also contribute to the problem whether it is consistent that every bounded function which is weakly measurable with respect to some Radon measure is Pettis integrable.
4
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A note on strictly positive Radon measures

100%
Plebanek G.
Colloquium Mathematicum
|
1996
|
tom 69
|
nr 2
187-192
5
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Nonseparable Radon measures and small compact spaces

100%
Plebanek G.
Fundamenta Mathematicae
|
1997
|
tom 153
|
nr 1
25-40
EN
We investigate the problem if every compact space K carrying a Radon measure of Maharam type κ can be continuously mapped onto the Tikhonov cube $[0, 1]^κ$ (κ being an uncountable cardinal). We show that for κ ≥ cf(κ) ≥ κ this holds if and only if κ is a precaliber of measure algebras. Assuming that there is a family of $ω_1$ null sets in $2^{ω1}$ such that every perfect set meets one of them, we construct a compact space showing that the answer to the above problem is "no" for κ = ω. We also give alternative proofs of two related results due to Kunen and van Mill [18].
6
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Convex Corson compacta and Radon measures

100%
Plebanek G.
Fundamenta Mathematicae
|
2002
|
tom 175
|
nr 2
143-154
EN
Assuming the continuum hypothesis, we show that (i) there is a compact convex subset L of $Σ(ℝ^{ω₁})$, and a probability Radon measure on L which has no separable support; (ii) there is a Corson compact space K, and a convex weak*-compact set M of Radon probability measures on K which has no $G_{δ}$-points.
7
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On Pettis integrals with separable range

100%
Plebanek G.
Colloquium Mathematicum
|
1993
|
tom 64
|
nr 1
71-78
EN
Several techniques have been developed to study Pettis integrability of weakly measurable functions with values in Banach spaces. As shown by M. Talagrand [Ta], it is fruitful to regard a weakly measurable mapping as a pointwise compact set of measurable functions - its Pettis integrability is then a purely measure-theoretic question of an appropriate continuity of a measure. On the other hand, properties of weakly measurable functions can be translated into the language of topological measure theory by means of weak Baire measures on Banach spaces. This approach, originated by G. A. Edgar [E1, E2], was remarkably developed by M. Talagrand. Following this idea, we show that the Pettis Integral Property of a Banach space E, together with the requirement of separability of E-valued Pettis integrals, is equivalent to the fact that every weak Baire measure on E is, in a certain weak sense, concentrated on a separable subspace. We base on a lemma which is a version of Talagrand's Lemma 5-1-2 from [Ta]. Our lemma easily yields a sequential completeness of the spaces of Grothendieck measures, a related result proved by Pallarés-Vera [PV]. We also present two results on Pettis integrability in the spaces of continuous functions.
8
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On asymptotic density and uniformly distributed sequences

64%
Frankiewicz R., Plebanek G.
Studia Mathematica
|
1996
|
tom 119
|
nr 1
17-26
EN
Assuming Martin's axiom we show that if X is a dyadic space of weight at most continuum then every Radon measure on X admits a uniformly distributed sequence. This answers a problem posed by Mercourakis [10]. Our proof is based on an auxiliary result concerning finitely additive measures on ω and asymptotic density.
9
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Countable tightness in the spaces of regular probability measures

64%
Plebanek G., Sobota D.
Fundamenta Mathematicae
|
2015
|
tom 229
|
nr 2
159-169
EN
We prove that if K is a compact space and the space P(K × K) of regular probability measures on K × K has countable tightness in its weak* topology, then L₁(μ) is separable for every μ ∈ P(K). It has been known that such a result is a consequence of Martin's axiom MA(ω₁). Our theorem has several consequences; in particular, it generalizes a theorem due to Bourgain and Todorčević on measures on Rosenthal compacta.
10
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Nonaccessible filters in measure algebras and functionals on $L^{∞}(λ)*$

38%
Frankiewicz R., Plebanek G.
Studia Mathematica
|
1994
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tom 108
|
nr 2
191-200
11
Content available remote

On Radon measures on first-countable spaces

32%
Plebanek G.
Fundamenta Mathematicae
|
1995
|
tom 148
|
nr 2
159-164
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