In: Mathematica Slovaca, vol. 57, no. 6
Josef Bukac
Detaily:
Rok, strany: 2007, 523 - 546
Kľúčové slová:
Hahn integral, nonmeasurable, outer measure, product space, seminorm, Minkowski inequality, completeness, convergence in outer measure,
class of sets with complete metric
O článku:
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.
Ako citovať:
ISO 690:
Bukac, J. 2007. Upper integral and its geometric meaning. In Mathematica Slovaca, vol. 57, no.6, pp. 523-546. 0139-9918.
APA:
Bukac, J. (2007). Upper integral and its geometric meaning. Mathematica Slovaca, 57(6), 523-546. 0139-9918.