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- W2056995512 abstract "Given † probability vector μ( X ) = (μ 1 ( X ), … μ k ( X )) of a finite number of components on a Borel class of sets X , we say that μ( X 0 ) has a diagonal value α if μ i ( X 0 ) = α for all i = 1, 2,…, K . J. Neyman(l), (2), (3) has proved that in the class of Borel sets of real numbers any non-atomic vector μ( X ) takes all diagonal values. A. Liapounoff has studied the full range of values of k -dimensional vector-valued measures and in two papers (4) he has proved that the range is closed and in the case of non-atomic measures the range is also convex. He also gave an example showing that neither of these results holds in the case of vectors of infinitely many components. A simplified proof of Liapounoff's results has been given by P. R. Halmos (5). In the present paper I study the range of values of probability vectors of infinitely many components. Various types of conditions are studied which are sufficient to imply that, for each ε > 0, 0 ≤ α ≤ 1, it is possible to find a set X such that" @default.
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- W2056995512 date "1961-10-01" @default.
- W2056995512 modified "2023-10-14" @default.
- W2056995512 title "On diagonal values of probability vectors of infinitely many components" @default.
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- W2056995512 doi "https://doi.org/10.1017/s0305004100035908" @default.
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