測度及びヒルバート空間の無限直積

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Kakutani [4] has pointed out that even if measures μ_i and ν_i on a same space Ω_i are equivalent with each other for every i=1,2, ... , their infinite product measures × μ_i and × ν_i on the product space × Ω_i are riot always equivalent, in fact, they are orthogonal under some conditions and he has given a criterion for these cases. By a small modification we may find his criterion has a close connection with the infinite direct product of H(μ_i) defined by J. von Neumann [5]. Originally J. von Neumann has constructed the infinite direct product of Hilbert spaces to utilize for definition of infinite direct product of algebras B_i of all bounded operators on Hilbert spaces H_i. However his definition is excessively formal and is not easy to understand it thoroughly. The purpose of this paper is, utilizing Kakutani's criterion, to draw out the fundamental characters of von Neumann's infinite direct product space. §1 is a preliminary section. In §2 Kakutani's theorem and in §3 von
茨城大学工学部の論文