Separability Of L～2(C,μ) And Its Haar Basis

Firstly, this paper has discussed the topological properties of the shift space, which is an important tool to study the fractal set, and itself is also a self-similar set. Then, by analyzing the properties of the Cantor set C, that is, the attractor of the iterated functionsystem{f₁ =1/3x,_f2 =1/3 (x+ 2),x∈R}, and the Cantor measureμ,that is, the invariant measure about the above iterated function system and the probability vector P = (1/2,1/2), this paper has proved that the space L^p (C,μ) (1≤p <∞) is separable by using the Weierstrass Proximity theory. As a special case, this conclusion also holds when p = 2.In addition, based on the analysis on the geometrical properties of the shiftspace (∑^∞,δ_r)and the discussion of the relations between the shift space (∑^∞,δ_r)and the Cantor set C, together with the theories and skills of fractal geometry, this paper has shown a Haar basis of L²(C,μ) clearly by way of construction. At last, this paper haspopularized the conclusion to the invariant set K of the similar contractive iterated function system which satisfies intensive separable condition and gained a Haar waveletbasis of L² (K,μ).