An Extension For The Packing Dimension And Measure Under The View Of Carathéodory Structure

In his writing"DIMENSION THEORY IN DYNAMICAL SYSTEMS: Contemporary Views and Applications",Professor Yakov B.Pesin enrolled a new concept of Carath(?)odory structure which extend into Carath(?)odory dimension and Carath(?)odory capacity.However,he did not let the packing dimension in,and he did not debate those three dimension at the abstract environment in detail.In this paper,we use the so-called"perfect cover"and"packing space"to extend the packing dimension(measure) via duality method and yield the Inequality:dim_C(E)≦dim(?)(E)=(?)_C(E)≦ (?)(E) which means the extension is well definite.As an application,we debate the abstract packing dimension of measure,and compare with the corresponding theorem in Carath(?)odory dimension and Caratheodory capacity.