Lineability of nowhere monotone measures (Q2258520)

Let \(X\) be a locally compact space without isolated points, and denote by \(\mathcal{M}(X)\) the Banach lattice of all real-valued Radon (Borel) measures on \(X\). The author calls \(\mu \in \mathcal{M}(X)\) nowhere monotone if \(\mu^{+}(G)>0\) and \(\mu^{-}(G)> 0\) for every nonempty open subset \(G\) of \(X\). The set \(\mathcal{N}(X)\) of such measures \(\mu\) is a dense \(G_\delta\) subset of \(\mathcal{M}(X)\) provided that \(X\) has a countable basis, which is a special case of Theorem 3.0.4 of the paper under review. Its main concern is, however, the situation where \(X = {\mathbb R}^d\) and \(\mathcal{M}({\mathbb R}^d)\) is replaced by \(\mathcal{A}\mathcal{C}_{{\lambda}_d}({\mathbb R}^d)\), its closed linear sublattice consisting of measures absolutely continuous with respect to the Lebesgue measure \({\lambda}_d\). Theorem 5.0.4 then says that there exists a dense linear subspace of \(\mathcal{A}\mathcal{C}_{{\lambda}_d}({\mathbb R}^d)\) of dimension \(\mathfrak c\) whose nonzero elements are all nowhere monotone. An essential ingredient of the proof is a recent result of \textit{L. Bernal-González} and \textit{M. Ordóñez Cabrera} [J. Funct. Anal. 266, No. 6, 3997--4025 (2014; Zbl 1298.46024), Theorem 2.3(c)]. Reviewer's remark: The subject of the paper is interesting and timely. The presentation has, however, many rather serious drawbacks, some of which are listed below. {\parindent=6mm \begin{itemize}\item[(1)] According to the author's nonstandard definition of absolute continuity of Radon measures, every real-valued Radon measure on \({\mathbb R}^d\) is absolutely continuous with respect to \({\lambda}_d\). \item[(2)] The proof of Lemma 3.0.3 is unnecessarily complicated. In fact, a simple application of the Hahn decomposition theorem would do. \item[(3)] Theorems 3.0.4 and 4.0.3 as well as their proofs are very close to some material of M. Kolář's bachelor thesis (Charles University in Prague, 2009; in Czech). The author does refer to that thesis, but only in connection with Definition 3.0.3. \item[(4)] The proof of Theorem 3.0.4 could be simplified by using the continuity of the mapping \({\mu}\mapsto{\mu}^+\) on \(\mathcal{M}(X)\) with respect to the norm topology. \item[(5)] The proof of Theorem 3.0.6 is unclear since the definition of \(I_n\) is wrong. \item[(6)] The crucial definition of differentiability of a real-valued Radon measure on \({\mathbb R}^d\) with respect to \({\lambda}_d\) is missing, which makes the material of Section 4 obscure. \item[(7)] The symbol \(\overline{spt}\) is not explained, which makes the arguments of Section 5 obscure. \item[(8)] The use of the Riesz representation theorem in the proof of Theorem 5.0.4 is pointless. \end{itemize}}