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

Richard D. Mabry¹
Fund. Math. 209 (2010), 95-113
doi:10.4064/fm209-2-1
Abstract: 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.

MSC (2010): Primary 28A12; Secondary 28A05.
Retrieve article in PDF (413.74 Kb)

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