A note on measurable functions
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 \(\mu\in {\mathcal M}\).
scientific article; zbMATH DE number 4000269