On additive absolutely nonmeasurable Sierpiński-Zygmund functions
Let \(E\) be a nonempty set, \(M\) be a set of real measures defined on \(E\) (measures defined on \(\sigma\)-algebras of subsets of \(E\)) and \(f: E\to\mathbb{R}\) be a function. The function \(f\) is called absolutely nonmeasurable with respect to \(M\) if \(f\) is nonmeasurable with respect to any measure from \(M\). Denote by \(M(E)\) the set of nonzero \(\sigma\)-finite diffused measures on \(E\) (i.e. vanishing on all singletons). The function \(f\) is called absolutely nonmeasurable if \(f\) is absolutely nonmeasurable with respect to \(M(E)\). A set \(X\) in \(\mathbb{R}\) is called a universal measure zero set if \(\mu^*(X)= 0\) for every \(\sigma\)-finite diffused Borel measure \(\mu\) on \(\mathbb{R},\mu^*\) being the outer measure associated with \(\mu\). The author uses the result that \(f\) is absolutely nonmeasurable if and only if the range of \(f\) is a universal measure zero subset of \(\mathbb{R}\) and \((\forall t\in\mathbb{R}) (\text{card}\{f^{-1}(t)\}\leq\omega),\) where \(\omega\) is a countable cardinal. A function \(f: \mathbb{R}\to\mathbb{R}\) is called a Vitali type function if \((\forall x\in \mathbb{R})(f(x)- x\in Q)\) and \(\text{ran}(f)\) is a selector of the partition \(\mathbb{R}/Q\). Then, if \(f\) is a Vitali type function, the Lebesgue measure \(\lambda\) can be extended to a measure \(\mu\) on \(\mathbb{R}\) such that \(f\) turns out to be measurable with respect to \(\mu\) (therefore, \(f\) is not absolutely nonmeasurable with respect to the class of all extensions of \(\lambda\)). It is known that \(f: \mathbb{R}\to\mathbb{R}\) is a solution of the Cauchy functional equation \((\forall(x,y)\in \mathbb{R}^2)(f(x+ y)= f(x)+ f(y))\) if and only if \(f\) is a homomorphism of the additive group \(\mathbb{R}\) into itself. A function \(f: \mathbb{R}\to\mathbb{R}\) is called a Sierpiński-Zygmund function if \((\forall Y\subset\mathbb{R})(\text{card\,}Y= c)(f/Y\text{ is not continuous on }Y),\) where \(c\) is the cardinality of the continuum. The author proves the following theorems: (1) If we assume Martin's axiom, there exists an injective, additive and absolutely nonmeasurable function \(f: \mathbb{R}\to\mathbb{R}\) such that \((\forall K\subset\mathbb{R})(\text{card\,}K= c)(f/K= l_K)\) and consequently \(f\) is not a Sierpiński-Zygmund function; (2) If \(G\) is any subset of \(\mathbb{R}\) which is a vector space on \(Q\) and has the cardinality \(c\), there exists an injective nontrivial solution of the Cauchy functional equation \(f: \mathbb{R}\to\mathbb{R}\) such that \(\text{ran}(f)\subset G\) and \(f\) is a Sierpiński-Zygmund function. (3) There exists an injective additive absolutely nonmeasurable Sierpiński-Zygmund function if we assume Martin's axiom.
scientific article; zbMATH DE number 5071514