On two theorems of Sierpi\'nski

TL;DR

This paper generalizes Sierpiński's theorems about infinite sets of reals containing disjoint subsets with equal measure or Baire property, providing stronger results and broader applicability.

Contribution

It introduces a general theorem extending Sierpiński's results, strengthening classical theorems regarding measure and Baire property in infinite sets.

Findings

Established a general theorem of this type.

Derived new corollaries strengthening classical results.

Extended Sierpiński's theorems to broader contexts.

Abstract

A theorem of Sierpi\'nski says that every infinite set Q of reals contains an infinite number of disjoint subsets whose outer Lebesgue measure is the same as that of Q. He also has a similar theorem involving the Baire property. We give a general theorem of this type and its corollaries, strengthening classical results.