On the Topological Dimension of Probability Measures

In 1932, L. S. Pontryagin and L. G. Schnirelmann proved a theorem according to which the greatest lower bound for the lower box dimensions of a metrizable compact space with respect to all metrics compatible with the topology equals the topological dimension of this compact space. This paper considers the greatest lower bound for the quantization dimensions of a Borel probability measure on a compact metrizable space with respect to all compatible metrics on this compact space. It is shown that, given probability measure on an interval of the real line, there exists a metric compatible with the topology of the interval for which the quantization dimension of this measure vanishes.