An intermediate-value theorem for the upper quantization dimension

Let μ be a Borel probability measure on Rd with compact support and Dr (μ) the upper quantization dimension of μ of order r. We prove, that for every t ∈ (dim∗p μ, dim∗Bμ], there exists a Borel probability measure ν with ν μ such that Dr (ν) = dim∗Bν =t. In addition, we give an example to show that the above intermediate-value property may fail in the open interval (dimp μ, dim∗p μ). Thus we get a complete description of the dimension set {Dr (ν): ν(Rd) = 1, ν μ}.