A moment inequality (Q1113169)

In a strictly measure-theoretic setting, the author proves the following result on the ratio of the moments of two related functions, f on a probability space (\(\Omega\),\(\mu)\) and w on [0,1]. Theorem: Let (\(\Omega\),\(\mu)\) be a probability space and \(\nu\) a nonatomic, Borel probability measure on [0,1]. Also, let w be a nonnegative, nondecreasing function on [0,1] with respect to \(\nu\) such that \[ 0<\int^{1}_{0}w^ n d\nu <\infty,\quad n<\infty, \] and f: \(\Omega\) \(\to {\mathbb{R}}^ a \)nonnegative \(\mu\)-measurable function with a nondecreasing quotient by w with respect to \(\nu\) (see the definitions given below). Then, for \(0<m\leq n\), we have \[ (M_{f,m}/M_{w,m})^{1/m} \leq (M_{f,n}/M_{w,n})^{1/n},\quad where\quad M_{f,m}=\int_{\Omega}f^ m d\mu \quad and\quad M_{w,m}=\int^{1}_{0}w^ m d\nu. \] The above inequality obviously reduces to the classical moment inequality \((M_{f,m})^{1/m} \leq (M_{f,n})^{1/n}\) for \(0<m\leq n<\infty\) when \(w\equiv 1\) and \(\nu\) \(\equiv Lebesgue\) measure. The following definitions are used. Definition 1: Let \(\nu\) be a totally finite, positive measure on [0,1]. An extended real-valued function w on [0,1] is said to be nondecreasing with respect to \(\nu\) if there is a \(\nu\)-measurable set D such that \(\nu (D)=\nu ([0,1])\) and w is a nondecreasing real-valued function on D. Definition 2: Let f: \(\Omega\) \(\to {\mathbb{R}}\) be a nonnegative \(\mu\)- measurable function on the probability space (\(\Omega\),\(\mu)\). Let w be a nonnegative extended real-valued function, nondecreasing with respect to a Borel probability measure \(\nu\) on [0,1]. Then f is said to have a nondecreasing quotient by w with respect to \(\nu\) if there is a probability space \((\Omega^*,\mu^*)\) and a nonnegative \((\mu^*\times \nu)\)-measurable function \(f_*\) on \(\Omega^*\times [0,1]\) which is equidistributed with f such that the following conditions are satisfied: There is a \(\mu^*\)-measurable set E and \(\nu\)-measurable set D \((\mu^*(E)=\nu (D)=1)\) such that (i) \(f_*(\omega^*,r)\) is \(\mu\)- measurable for any \(r\in D\), (ii) \(f_*(\omega^*,r)/w(r)\) is real- valued and nondecreasing for any \(\omega^*\in E.\) Definition 3: For two probability spaces \((\Omega_ i,\mu_ i)\), \(i=1,2\), and a \(\mu_ i\)-measurable, nonnegative, real-valued function \(f_ i\) on \(\Omega_ i\), we say that \(f_ 1\) and \(f_ 2\) are equidistributed if \[ \mu_ 1(\{\omega_ 1:f_ 1(\omega_ 1)>Y\})=\mu_ 2(\{\omega_ 2:f_ 2(\omega_ 2)>Y\}) \] for any \(Y\in R\).