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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm209-2-1,
author = {Richard D. Mabry},
title = {Stretched shadings and a Banach measure that is not scale-invariant},
journal = {Fundamenta Mathematicae},
volume = {209},
year = {2010},
pages = {95-113},
zbl = {1213.28001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm209-2-1}
}
Richard D. Mabry. Stretched shadings and a Banach measure that is not scale-invariant. Fundamenta Mathematicae, Tome 209 (2010) pp. 95-113. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm209-2-1/