How to characterize some properties of measurable functions (Q2717017)

Let \((\Omega,\mathcal F)\) be a measurable space (where the elements of \(\mathcal F\) are referred to as measurable sets). The optimal measure is defined as a set function \(p:\mathcal F\to [0,1]\) fulfilling the following axioms:NEWLINENEWLINENEWLINE\(p(\emptyset)=0\) and \(p(\Omega)=1\);NEWLINENEWLINENEWLINE\(p(E_1\cup E_2)=\max(p(E_1),p(E_2))\) for all measurable sets \(E_1\) and \(E_2\); NEWLINENEWLINENEWLINE\(p\left(\bigcap\limits_{n=1}^\infty E_n\right)=\lim\limits_{n\to\infty} p(E_n)\) for every decreasing sequence of measurable sets \(E_n\). NEWLINENEWLINENEWLINELet \(s=\sum\limits_{i=1}^n b_i\chi(B_i)\) be a nonnegative measurable simple function, where \(B_i\) are measurable sets, \(\{B_1,\dots,B_n\}\) is a partition of \(\Omega\) and \(\chi(B)\) is the indicator function of the set \(B\). The \textit{optimal average} \(\int_\Omega s dp\) of \(s\) is defined as \(\max\limits_{1\leq i\leq n}b_i p(B_i)\). This quantity does not depend on the decomposition of \(s\). The optimal average \(\int_\Omega\left|f\right|dp\) of a measurable function \(f\) is defined by \(\sup \int_\Omega s dp\), where the supremum is taken over all measurable simple functions \(s\) for which \(0\leq s\leq\left|f\right|\). NEWLINENEWLINENEWLINEThe author gives in terms of optimal average necessary and sufficient conditions in order that a measurable function \(f\) be bounded. He also studies conditions (also in terms of optimal average) in order that a sequence of measurable functions be uniformly, pointwise, discretely, equally or quasi-uniformly convergent. NEWLINENEWLINENEWLINERemark: The definition of quasi-uniformly convergent sequence of functions given by author is confusing.