## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Lineability of Nowhere Monotone Measures

Petr Petràček

#### 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.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 21, Number 5 (2014), 873-885.

Dates
First available in Project Euclid: 1 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1420071859

Digital Object Identifier
doi:10.36045/bbms/1420071859

Mathematical Reviews number (MathSciNet)
MR3298483

Zentralblatt MATH identifier
1326.46023

#### Citation

Petràček, Petr. Lineability of Nowhere Monotone Measures. Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 5, 873--885. doi:10.36045/bbms/1420071859. https://projecteuclid.org/euclid.bbms/1420071859