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Dai_FeinbergG

# Dai_FeinbergG - ON MAXIMAL RANGES OF VECTOR MEASURES FOR...

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ON MAXIMAL RANGES OF VECTOR MEASURES FOR SUBSETS AND PURIFICATION OF TRANSITION PROBABILITIES PENG DAI AND EUGENE A. FEINBERG Abstract. Consider a measurable space with an atomless finite vector mea- sure. This measure defines a mapping of the σ -field into an Euclidean space. According to the Lyapunov convexity theorem, the range of this mapping is a convex compactum. Similar ranges are also defined for measurable subsets of the space. Two subsets with the same vector measure may have different ranges. We investigate the question whether, among all the subsets having the same given vector measure, there always exists a set with the maximal range of the vector measure. The answer to this question is positive for two- dimensional vector measures and negative for higher dimensions. We use the existence of maximal ranges to strengthen the Dvoretzky-Wald-Wolfowitz pu- rification theorem for the case of two measures. 1. Introduction Let ( X, F ) be a measurable space and μ = ( μ 1 , ..., μ m ), m = 1 , 2 , . . . , be a finite atomless vector measure on it. We recall that a measure ν is called atomless if for each Z ∈ F , such that ν ( Z ) > 0, there exists Z 0 ∈ F such that Z 0 Z and 0 < ν ( Z 0 ) < ν ( Z ). A vector measure μ = ( μ 1 , ..., μ m ), is called finite and atomless if each measure μ i , i = 1 . . . m , is finite and atomless. For each Y ∈ F consider the range R μ ( Y ) = { μ ( Z ) : Z ∈ F , Z Y } of the vector measures of all its measurable subsets Z . According to the Lyapunov convexity theorem [8], the sets R μ ( Y ) are convex compactums in R m . For a review on this theorem and its applications see [9]. In this paper we study whether for any p R μ ( X ), the set of all ranges { R μ ( Y ) : μ ( Y ) = p, Y ∈ F} contains a maximal element. In other words, is it always true that for any p R μ ( X ) there exists a subset Y * ∈ F such that μ ( Y * ) = p and R μ ( Y * ) R μ ( Y ) for any Y ∈ F with μ ( Y ) = p ? We show that the answer is positive when m = 2 and negative when m > 2. Furthermore, for m = 2, this maximal range can be constructed by a simple geometric transformation of R μ ( X ) . In addition to the maximal range, it is possible to consider a minimal range. For q R μ ( X ), the set M * ∈ F with μ ( M * ) = q has minimal range corresponding to q if R μ ( M ) R μ ( M * ) for any M ∈ F with μ ( M ) = q . We show that a set has a maximal range corresponding to p if and only if its complement has a minimal range Date : June 24, 2010. 2010 Mathematics Subject Classification. Primary 60A10, 28A10. Key words and phrases. Lyapunov convexity theorem, maximal subset, purification of transi- tion probabilities. This research was partially supported by NSF grants CMMI-0900206 and CMMI-0928490. 1

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2 PENG DAI AND EUGENE A. FEINBERG corresponding to μ ( X ) - p . Therefore, minimal ranges also exist for dimension two and they may not exist for higher dimensions.
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