A note on measurable functions
type: Item
instance of: scholarly article
title: A note on measurable functions
zbMATH Open document ID: 0617.28003
DOI: 10.1007/BF01189863
author: Wolfgang Filter
published in: Archiv der Mathematik
publication date: 1986-01-01T00:00:00Z
review text: Let X be a non-empty set, \({\mathcal R}^ a \delta\)-ring of X and \({\mathcal M}^ a \)band of the Dedekind complete Riesz space of all real-valued measures on \({\mathcal R}\). For \(f\in {\bar {\mathbb{R}}}^ X\) let \[ {\mathcal M}(f):=\{\mu \in {\mathcal M}| f\in {\mathcal L}^ 1(\mu)\}. \] The function f is called \(\mu\)-measurable iff for every \(\alpha\in {\mathbb{R}}\) and \(C\in {\mathcal R}\) the characteristic function of \(\{x\in C| f(x)>\alpha \}\) is in \({\mathcal L}^ 1(\mu).\) The main result of this note is the following equivalence for \(f\in {\bar {\mathbb{R}}}^ X:\) \({\mathcal M}(f)\) is order dense in \({\mathcal M}\Leftrightarrow\) f is \(\mu\)-measurable and \(\mu\)-a.e. finite for every \(\mu\in {\mathcal M}\).
Mathematics Subject Classification ID: 28A20
Mathematics Subject Classification ID: 28A12
zbMATH DE Number: 4000269
zbMATH Keywords: measurable functions
zbMATH Keywords: band
zbMATH Keywords: Dedekind complete Riesz space
MaRDI profile type: Publication
cites work: Duality in measure theory
cites work: Q3679426
full work available at URL: https://doi.org/10.1007/bf01189863
OpenAlex ID: W2054030933
description: scientific article; zbMATH DE number 4000269