On two theorems of Sierpiński

Abstract

A theorem of Sierpiński 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.

In the 1st replacement Proposition 3.4, Corollary 3.5 and 3.6 have been added. In the 2nd replacement subsection 4.2 in the Appendix (= Section 4) has been added. Next, a few improvement in the presentation were done, reference [12] was added, and a correction in the proof of Proposition 4.1 was introduced. Presently, historical comments are slightly modified