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december 2014 Lineability of Nowhere Monotone Measures
Petr Petràček
Bull. Belg. Math. Soc. Simon Stevin 21(5): 873-885 (december 2014). DOI: 10.36045/bbms/1420071859

Abstract

We give a sufficient condition for the set of nowhere monotone measures to be a residual $G_{\delta}$ set in a subspace of signed Radon measures on a locally compact second-countable Hausdorff space with no isolated points. We prove that the set of nowhere monotone signed Radon measures on a $d$-dimensional real space $\mathbb{R}^{d}$ is lineable. More specifically, we prove that there exists a vector space of dimension $\mathfrak{c}$ (the cardinality of the continuum) of signed Radon measures on $\mathbb{R}^{d}$ every non-zero element of which is a nowhere monotone measure that is almost everywhere differentiable with respect to the $d$-dimensional Lebesgue measure. Using this result we show that the set of these measures is even maximal dense-lineable in the space of bounded signed Radon measures on $\mathbb{R}^{d}$ that are almost everywhere differentiable with respect to the $d$-dimensional Lebesgue measure.

Citation

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Petr Petràček. "Lineability of Nowhere Monotone Measures." Bull. Belg. Math. Soc. Simon Stevin 21 (5) 873 - 885, december 2014. https://doi.org/10.36045/bbms/1420071859

Information

Published: december 2014
First available in Project Euclid: 1 January 2015

zbMATH: 1326.46023
MathSciNet: MR3298483
Digital Object Identifier: 10.36045/bbms/1420071859

Subjects:
Primary: 28A33, 46E27

Rights: Copyright © 2014 The Belgian Mathematical Society

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Vol.21 • No. 5 • december 2014
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