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

Richard D. Mabry¹

¹ Department of Mathematics Louisiana State University in Shreveport Shreveport, LA 71115-2399, U.S.A.

Fundamenta Mathematicae, Tome 209 (2010) no. 2, pp. 95-113

Résumé

It is shown that if $A\subset\mathbb R$ 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 $A\subset\mathbb R$ has constant $\mu$-shade with respect to some fixed Banach measure $\mu$, then the same need not be true of a similarity transformation of $A$ with respect to $\mu$. But even if it is, the $\mu$-shade might be changed by the transformation. To prove such a $\mu$ 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.

Détail

DOI : 10.4064/fm209-2-1

Keywords: shown subset mathbb has constant shade respect banach measures similarity transformation shade changed transformation other subset mathbb has constant mu shade respect fixed banach measure similarity transformation respect even mu shade might changed transformation prove exists hamel basis weak closure properties respect multiplication construct sets convenient scaling properties notion shade almost invariance introduced aiding construction variety banach measures particular scale invariant