## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

### Lineability of Nowhere Monotone Measures

#### 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

**Subjects**

Primary: 46E27: Spaces of measures [See also 28A33, 46Gxx] 28A33: Spaces of measures, convergence of measures [See also 46E27, 60Bxx]

**Keywords**

lineability nowhere monotone measures absolutely continuous measures spaces with humps

#### 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