A note on stochastic operators on \(L^ 1\)-spaces and convex functions
type: Item
instance of: scholarly article
title: A note on stochastic operators on \(L^ 1\)-spaces and convex functions
zbMATH Open document ID: 0644.60002
DOI: 10.1016/0022-247X(88)90332-0
author: Peter M. Alberti
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: Main result: Let (X,m) denote a localizable measure space and call any \(f\in L_ 1(X,m)\) satisfying \(f\geq 0\) m-a.e. and \(\int f d\mu =1\) a probability distribution over (X,m). Then for any n-tuples \((f_ 1,...,f_ n)\), \((g_ 1,...,g_ n)\) of probability distributions over (X,m) there exists some \(T:L_ 1(X,m)\to L_ 1(X,m)\) satisfying T(f)\(\geq 0\) and \(\| T(f)\|_ 1=\| f\|_ 1\) for any \(f\in L_ 1(X,m)\), \(f\geq 0\) m-a.e., such that \(T(f_ j)=g_ j\), \(j=1,...,n\), is valid, if any only if for any \(F:{\mathbb{R}}_+\) \(n\to {\mathbb{R}}\), which is nonnegative, homogeneous of degree one, and subadditive, the inequalities \(\int f(g_ j)dm\leq \int F(f_ j)dm\), \(j=1,...,n\), hold true.
Mathematics Subject Classification ID: 60A99
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 60E10
zbMATH DE Number: 4050618
zbMATH Keywords: n-tupel problem for probability distributions
zbMATH Keywords: localizable measure space
zbMATH Keywords: inequalities
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4050618