Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A multivariate generalization of a theorem of R. H. Farrell

Author: D. Plachky
Journal: Proc. Amer. Math. Soc. 113 (1991), 163-165
MSC: Primary 28A60; Secondary 28A25
MathSciNet review: 1052873
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Abstract: Let $ (\Omega ,\mathfrak{S},\mu )$ denote a finite measure space and $ {T_j}:\Omega \to [{a_j},{b_j}](\mathfrak{S},\mathfrak{B} \cap [{a_j},{b_j}])$-measurable functions, $ j = 1, \ldots ,n$. Then the algebra of functions generated by $ 1,{T_1}, \ldots ,{T_n}$ is a dense subset of $ {\mathcal{L}_1}(\Omega ,\mathfrak{S},P)$ if and only if for any $ A \in \mathfrak{S}$ there exists some $ B \in {({T_1}, \ldots ,{T_n})^{ - 1}}({\mathfrak{B}^n} \cap [{a_1},{b_1}] \times \cdots \times [{a_n},{b_n}])$, such that $ \mu (A\Delta B) = 0$ is valid. In particular, this condition is satisfied if $ (\Omega ,\mathfrak{S})$ is a Blackwell space and $ ({T_1}, \ldots ,{T_n}):\Omega \to [{a_1},{b_1}] \times \cdots \times [{a_n},{b_n}]$ is in addition one-to-one.

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Article copyright: © Copyright 1991 American Mathematical Society