Upper integral and its geometric meaning
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11150%2F07%3A00004088" target="_blank" >RIV/00216208:11150/07:00004088 - isvavai.cz</a>
Result on the web
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DOI - Digital Object Identifier
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Alternative languages
Result language
angličtina
Original language name
Upper integral and its geometric meaning
Original language description
The Hahn definition of the integral is recalled, the requirement of measurability of the integrand omitted. Both the upper anl lower integrals comply with this definition and so does any measurable function between them. The outer product measure of thehypograph 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 andd 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 thesymmetric difference of two sets gives us a complete metric space of classes of subsets.
Czech name
Horní integrál a jeho geometrický význam
Czech description
Hahnova definice integrálu je znovu otevřena, požadavek na měřitelnost integrálů je opuštěn. Oba horní i dolní limity integrálu jsou v souladu s definicí a tak tomu je i pro všechny měřitelné funkce mezi těmito hranicemi. Hodnota hypografu nonnegativní vázané neměřitelné funkce je rovna hornímu integrálu, který je roven jednomu z Fan integrálů. Normalita pro neměřitelné funkce je definována horním integrálem. Konvergence v tomto prostoru implikuje konvergenci ve vnějším prostoru.
Classification
Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Result continuities
Project
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Continuities
S - Specificky vyzkum na vysokych skolach
Others
Publication year
2007
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Mathematica Slovaca
ISSN
0139-9918
e-ISSN
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Volume of the periodical
57
Issue of the periodical within the volume
6
Country of publishing house
PL - POLAND
Number of pages
24
Pages from-to
523-546
UT code for WoS article
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EID of the result in the Scopus database
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