Some Properties On Hewitt-Stromberg Measures

Fractal dimensions play an important role in the study of fractal geometry and some of the frequently used fractal dimensions can be deduced by the corresponding measures.The Hewitt-Stromberg measure is an outer measure introducted by Hewitt and Stromberg in 1965.It is strongly related to Hausdorff measure and packing measure,so some of their properties are similar.In this thesis,we systematically discuss its properties and the properties of the dimension deduced by it.In particular,we get two main results.First,we obtain an estimation formula for the lower Hewitt-Stomberg measure of the product set in the general metric space,which generalizes the result of Guizani et al.(Ann.Mat.Pura Appl.(2021))from Euclian spaces to the general metric spaces.This is achieved by constructing a measure,which is equivalent to the lower Hewitt-Stromberg measure in the metric space.Secondly,we obtain a theorem for the existence of finite measure subset of lower Hewitt-Stremberg measure,which extends the classical theorem for the existence of finite measure subset of Hausdorff measure.That is,in an uncountable complete separable metric space,there exists a compact perfect subset and a dimension function such that the lower Hewitt-Stromberg measure of compact perfect subset is finite.This prove is achieved by constructing a set similar to the middle third Cantor set in an uncountable complete separable metric space.Finally,we give some unsolved problems related to the Hewitt-Stromberg measure.