Affiliation:
1. Faculty of Pure and Applied Mathematics , Wrocław , Poland
Abstract
Abstract
Let (X, ℱ) be a measurable space with a nonatomic vector measure µ =(µ
1, µ
2). Denote by R(Y) the subrange R(Y)= {µ(Z): Z ∈ ℱ, Z ⊆ Y }. For a given p ∈ µ(ℱ) consider a family of measurable subsets ℱ
p
= {Z ∈ ℱ : µ(Z)= p}. Dai and Feinberg proved the existence of a maximal subset Z* ∈ Fp
having the maximal subrange R(Z*) and also a minimal subset M* ∈ ℱ
p
with the minimal subrange R(M*). We present a method of obtaining the maximal and the minimal subsets. Hence, we get simple proofs of the results of Dai and Feinberg.
Reference7 articles.
1. [1] BIANCHINI, S.—CERF, R.—MARICONDA, C.: Two-dimensional zonoids and Chebyshev measures, J. Math. Anal. Appl. 211 (1997), 512–526.
2. [2] CANDELORO, D.—MARTELLOTTI, A.: Geometric properties of the range of two-dimensional quasi-measures with respect to the Radon-Nikodym property, Adv. Math. 93 (1992), 9–24.
3. [3] DAI, P .—FEINBERG, A.: On maximal ranges of vector measures for subsets and purification of transition probabilities, Proc. Amer. Math. Soc. 139 (2011), 4497–4511.
4. [4] LEGUT, J.—WILCZYŃSKI, M.: How to obtain a range of a nonatomic vector measure in ℝ2, J. Math. Anal. Appl. 394 (2012), 102–111.
5. [5] LEHMANN, E. L.—ROMANO, J. P.: Testing Statistical Hypotheses. Springer Science + Business Media. Inc., New York, 1971.