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.