On the functor of probability measures and quantization dimensions

The quantization dimensions of the probability measure given on the metric compact coincide with the dimensions of the finite approximation for the probability measure functor. Some functorial properties of quantization dimensions are established. It is shown that for any $b>0$ there exists a metric compact $X_b$ of capacitive dimension $\mathrm{dim}_{\mathrm{B}}X_b = b$ on which there are probability measures with support equal to $X$ whose quantization dimension takes all possible values from the interval $[0, b]$.