Net Measure's Qualities And Applications

In this dissertation, a detailed definition of n-net measure is presented, and some of its basic properties and expressions are researched. Firstly, this paper gives an brief introduction to the fractal geometry,measure theory,the measure of fractals, ect. Secondly, it has given explicit definitions of net measure and regular net measure. Thirdly, it has studied some concrete properties of the net measure, for example: outer measure,distant measure,move invariance. It has a brief account of the existing research on the net measure, including the net measure of Moran sets, the net measure of Cantor sets and its applications and such like. The net measure is relatively easier to calculate because it meets with the principle of mass distribution , its cover has certain regularity and it avoid the arbitrary and overlapping of Hausdorff measure. With this tool, it is easy to estimate the size of fractal sets and it is also a new way to calculate the fractal sets' upper bound of Hausdorff measure, which gives a relatively easier way to study Hausdorff dimension of fractal sets. Finally, as a kind of application, we calculate the net measure of such fractal sets : third Cantor set, Sierpinski carpet, Sierpinski gasket ,etc. At present, research in this area is still in preliminary and we have little research on the move invariability and measurement result under different systems. So it needs us to go further. Anyhow, The net measure has the vital significance to the fractal geometry research and this article has farther conducted some research to the net measure theory.