A note on stochastic operators on \(L^ 1\)-spaces and convex functions
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.
n-tupel problem for probability distributions
scientific article; zbMATH DE number 4050618
scientific article