Sets which are well-distributed and invariant relative to all isometry invariant total extensions of Lebesgue measure (Q1175317)

Let \(\mu\) be an isometry invariant, finitely additive set function defined for all subsets of \(\mathbb{R}\) such that \(\mu(E)=\lambda(E)\) for Lebesgue measurable sets \(E\). The author shows the fact, rather unexpected in view of the remarks of \textit{A. Simoson} [Am. Math. Mon. 89, 114-116 (1982; Zbl 0496.28004)], that, for \(0<\alpha<1\), there exist sets \(A\subset \mathbb{R}\) with the property \(\mu(A\cap E)=\alpha \lambda(E)\) for all Lebesgue measurable sets \(E\). More generally, if \(f: \mathbb{R}\to[0,1 ]\) is a given continuous function, there exists a set \(F\subset \mathbb{R}\) such that \(\lim[\mu(F\cap J(x))/\mu(J(x))]=f(x)\) for \(x\in\mathbb{R}\) and the limit being taken for intervals \(J(x)\) containing \(x\) and with \(\mu(J(x))\to 0\). Many examples, applications and unsolved problems are given.