Un problème sur une classe de mesures non-lebesguinnes. (A problem on a class of non-Lebesgue measures) (Q1078346)

In a previous note (see the review above) \textit{T. Amaducci} and the author introduce a class \({\mathbb{M}}_{\mu}({\mathbb{P}})\) of non-Lebesgue and \(\sigma\)-additive measures on a bounded, nowhere dense, perfect subset \({\mathbb{P}}\) (even of zero Lebesgue measure) of the real axis. In the present work the author sets the following problem for the class \({\mathbb{M}}_{\mu}({\mathbb{P}}):\) Does it exist a non empty class \(\Sigma\) of subsets \(E\subset [a,b]\) such that, for any sequence \(\{{\mathbb{P}}_ n\}\) of nowhere dense, perfect subsets of \([a,b]\) for which \(\lim_{n,\infty}m({\mathbb{P}}_ n)=m([a,b]),\) there is a sequence \(\{\mu_ n\}\) of measures \(\mu_ n\in {\mathbb{M}}_{\mu}({\mathbb{P}}_ n)\) such that \(\lim_{n,\infty}\mu^*_ n(E)=m^*(E)\) for any \(E\in \Sigma ?\) (Here, as usually, one denotes by m, \(\lambda^*\) respectively the Lebesgue measure and the outer measure relative to the measure \(\lambda\).) Moreover, by giving the construction of a suitable sequence \(\{\mu_ n\}\) of measures, the author shows that the class of Peano- Jordan measurable subsets is a solution of the above problem, but it is not a maximal class-solution.