Haar測度と平均値について

概要

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The object of this paper is to make use of the mean of the continuos function in the construction of the Haar's measure for a compact topological group. Let G be a compact topological group and C_A the indicator of a subset A of G. Let T denote the Baire function class and B the Borel family of the sets. Now, let M(f) denote the mean of f for any function f∈T, where the "mean" of f is the limit of the sequence {M(f_n)}, as f_n→f (n→∝, f_n belong the preceding class of f). Then we define the Haar's measure μ* as the following formula; μ*(A)=inf M(f){C_A≦f, f∈T_1} for any A⊂G, where T_1 is the Baire function family of the first class. Hence ∫f(x)dμ(x)=M(f), for any f∈T specially ∫1dμ(x)=M(1)=1 From the uniqueness of the Haar's measure, the Haar's measure on G be constructed by the method as described above.
岐阜工業高等専門学校の論文
1970-03-25