On the range of the quantization dimension of probability measures on a metric compactum

The quantization dimension of a probability measure on a metric compactum $X$ does not exceed the box dimension of the support of the measure. We prove the following intermediate value theorem for the upper quantization dimension: If $X$ is a metric compact space whose upper box dimension is equal to $a\leq\infty$ then for every real $b$ such that $0\leq b\leq a $ there exists a probability measure on $X$ whose support is $X$ and whose upper quantization dimension is $b$.