{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern