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On Marczewski-Burstin Representations Of Certain Algebras Of Sets, Krzysztof Ciesielski
On Marczewski-Burstin Representations Of Certain Algebras Of Sets, Krzysztof Ciesielski
Faculty & Staff Scholarship
We show that the Generalized Continuum Hypothesis GCH (its appropriate part) implies that many natural algebras on R, including the algebra B of Borel sets and the interval algebra S, are outer Marczewski-Burstin representable by families of non-Borel sets. Also we construct, assuming again an appropriate part of GCH, that there are algebras on R which are not MB-representable. We prove that some algebras (including B and S) are not inner MB-representable. We give examples of algebras which are inner and outer MB-representable, or are inner but not outer MB-representable.
Measure Zero Sets Whose Algebraic Sum Is Non-Measurable, Krzysztof Ciesielski
Measure Zero Sets Whose Algebraic Sum Is Non-Measurable, Krzysztof Ciesielski
Faculty & Staff Scholarship
In this note we will show that for every natural number n > 0 there exists an S ⊂ [0, 1] such that its n-th algebraic sum nS = S + ··· + S is a nowhere dense measure zero set,but its n+ 1-st algebraic sum nS +S is neither measurable nor it has the Baire property. In addition,the set S will be also a Hamel base,that is,a linear base of R over Q.
Measure Zero Sets With Non-Measurable Sum, Krzysztof Ciesielski
Measure Zero Sets With Non-Measurable Sum, Krzysztof Ciesielski
Faculty & Staff Scholarship
For any subset C of R there is a subset A of C such that A+A has inner measure zero and outer measure the same as C+C. Also, there is a subset A of the Cantor middle third set such that A+A is Bernstein in [0,2]. On the other hand there is a perfect set C such that C+C is an interval I and there is no subset A of C with A+A Bernstein in I.