Upper integral and its geometric meaning

Mathematica Slovaca

Josef Bukac¹ 

关键词: Hahn integral;    nonmeasurable;    outer measure;    product space;    seminorm;    Minkowski inequality;    completeness;    convergence in outer measure;    class of sets with complete metric;   

DOI  :  10.2478/s12175-007-0044-1

学科分类：数学（综合）

来源: Slovenska Akademia Vied * Matematicky Ustav / Slovak Academy of Sciences, Mathematical Institute

【 摘 要 】

The Hahn definition of the integral is recalled, the requirement of measurability of the integrand omitted. Both the upper and lower integrals comply with this definition and so does any measurable function between them.The outer product measure of the hypograph of a nonnegative bounded nonmeasurable function is equal to the upper integral which is equal to one of the Fan integrals. The outer measure of the graph of a bounded nonmeasurable function is equal to the difference between the upper and lower integrals.A norm for not necessarily measurable functions is defined with the upper integral. The linear space with this norm is complete. The convergence in this space implies the convergence in outer measure. The distance as an outer measure of the symmetric difference of two sets gives us a complete metric space of classes of subsets.

【 授权许可】

Unknown