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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.