INSTITUTE OF MATHEMATICS · POLISH ACADEMY OF SCIENCES
FUNDAMENTA
MATHEMATICAE
ISSN: 0016-2736(p) 1730-6329(e)
Stretched shadings and a Banach measure that is not scale-invariant
Richard D. Mabry^{1} 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.