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Extreme topological measures

100%
Butler S.
Fundamenta Mathematicae
|
2006
|
tom 192
|
nr 2
141-153
EN
It has been an open question since 1997 whether, and under what assumptions on the underlying space, extreme topological measures are dense in the set of all topological measures on the space. The present paper answers this question. The main result implies that extreme topological measures are dense on a variety of spaces, including spheres, balls and projective planes.
2
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Density in the space of topological measures

100%
Butler S.
Fundamenta Mathematicae
|
2002
|
tom 174
|
nr 3
239-251
EN
Topological measures (formerly "quasi-measures") are set functions that generalize measures and correspond to certain non-linear functionals on the space of continuous functions. The goal of this paper is to consider relationships between various families of topological measures on a given space. In particular, we prove density theorems involving classes of simple, representable, extreme topological measures and measures, hence giving a way of approximating various topological measures by members of different classes.
3
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Moving averages

63%
Butler S., Rosenblatt J.
Colloquium Mathematicum
|
2008
|
tom 113
|
nr 2
251-266
EN
In ergodic theory, certain sequences of averages ${A_k f}$ may not converge almost everywhere for all f ∈ L¹(X), but a sufficiently rapidly growing subsequence ${A_{m_k} f}$ of these averages will be well behaved for all f. The order of growth of this subsequence that is sufficient is often hyperexponential, but not necessarily so. For example, if the averages are $A_k f(x) = 1/(2^k) ∑_{j=4^k+1}^{4^k+2^k} f(T^jx)$, then the subsequence $A_{k²} f$ will not be pointwise good even on $L^∞$, but the subsequence $A_{2^k} f$ will be pointwise good on L¹. Understanding when the hyperexponential rate of growth of the subsequence is required, and giving simple criteria for this, is the subject that we want to address here. We give a fairly simple description of a wide class of averaging operators for which this rate of growth can be seen to be necessary.
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