Stretched shadings and a Banach measure that is not scale-invariant

• Autorzy: Richard D. Mabry
• Języki publikacji: EN

Abstrakty

EN

It is shown that if A ⊂ ℝ has the same constant shade with respect to all Banach measures, then the same is true of any similarity transformation of A and the shade is not changed by the transformation. On the other hand, if A ⊂ ℝ has constant μ-shade with respect to some fixed Banach measure μ, then the same need not be true of a similarity transformation of A with respect to μ. But even if it is, the μ-shade might be changed by the transformation. To prove such a μ exists, a Hamel basis with some weak closure properties with respect to multiplication is used to construct sets with some convenient scaling properties. The notion of shade-almost invariance is introduced, aiding in the construction of a variety of Banach measures, in particular, one that is not scale-invariant.

Kategorie tematyczne

• 28A12: Contents, measures, outer measures, capacities
• 28A05: Classes of sets (Borel fields, σ -rings, etc.), measurable sets, Suslin sets, analytic sets

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2010
• Tom: 209
• Numer: 2
• Strony: 95-113

Daty

wydano

2010

Twórcy

autor

Richard D. Mabry

• Department of Mathematics, Louisiana State University in Shreveport, Shreveport, LA 71115-2399, U.S.A.

Identyfikatory

DOI

10.4064/fm209-2-1