Open AccessHow to Obtain Maximal and Minimal Subranges of Two-Dimensional Vector Measures
Author(s) -
Jerzy Legut,
Maciej Wilczyński
Publication year - 2019
Publication title -
tatra mountains mathematical publications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.171
H-Index - 12
eISSN - 1338-9750
pISSN - 1210-3195
DOI - 10.2478/tmmp-2019-0022
Subject(s) - mathematics , measure (data warehouse) , combinatorics , space (punctuation) , simple (philosophy) , mathematical proof , discrete mathematics , geometry , computer science , operating system , philosophy , epistemology , database
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* ∈ F p 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.