On the Relationship between the Dimension of the Lebesgue-Stieltjes Measure and the Rate of Approximation of a Function by Step Functions

摘要

The relationship between the rate of approximation of a monotone function by step functions (with an increasing number of values) and the Hausdorff dimension of the corresponding Lebesgue-Stieltjes measure is studied. An upper bound on the dimension is found in terms of the approximation rate, and it is shown that a lower bound cannot be constructed in these terms.