An Extension of the 1-Dim Lebesgue Integral of a Product of Two Functions
In this paper, our main aim is to present a reasonable extension of the 1-dim Lebesgue integral of the product of two functions, in case this Lebesgue integral does not exist (i.e., the integrals of its negative and positive parts are both <i>∞</i>). This extension works fine quite gener...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-07-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/11/15/3341 |
| _version_ | 1827731170455977984 |
|---|---|
| author | Clara Carlota António Ornelas |
| author_facet | Clara Carlota António Ornelas |
| author_sort | Clara Carlota |
| collection | DOAJ |
| description | In this paper, our main aim is to present a reasonable extension of the 1-dim Lebesgue integral of the product of two functions, in case this Lebesgue integral does not exist (i.e., the integrals of its negative and positive parts are both <i>∞</i>). This extension works fine quite generally, as shown by several examples, and it is based on general hypotheses guaranteeing the sign of the integral (in the sense of being necessarily <0 or =0 or else >0), without computing its actual value. For this purpose, our method provides much more precise results than the Lebesgue–Stieltjes integration by parts. |
| first_indexed | 2024-03-11T00:22:41Z |
| format | Article |
| id | doaj.art-a040aa391f904cea9cb63b57bfad2acc |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T00:22:41Z |
| publishDate | 2023-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-a040aa391f904cea9cb63b57bfad2acc2023-11-18T23:15:21ZengMDPI AGMathematics2227-73902023-07-011115334110.3390/math11153341An Extension of the 1-Dim Lebesgue Integral of a Product of Two FunctionsClara Carlota0António Ornelas1Departamento de Matemática, CIMA, Universidade de Évora, 7000-671 Évora, PortugalDepartamento de Matemática, CIMA, Universidade de Évora, 7000-671 Évora, PortugalIn this paper, our main aim is to present a reasonable extension of the 1-dim Lebesgue integral of the product of two functions, in case this Lebesgue integral does not exist (i.e., the integrals of its negative and positive parts are both <i>∞</i>). This extension works fine quite generally, as shown by several examples, and it is based on general hypotheses guaranteeing the sign of the integral (in the sense of being necessarily <0 or =0 or else >0), without computing its actual value. For this purpose, our method provides much more precise results than the Lebesgue–Stieltjes integration by parts.https://www.mdpi.com/2227-7390/11/15/3341extension of the 1-dim Lebesgue integral of a productintegral inequalitiesLebesgue–Stieltjes integration by parts |
| spellingShingle | Clara Carlota António Ornelas An Extension of the 1-Dim Lebesgue Integral of a Product of Two Functions Mathematics extension of the 1-dim Lebesgue integral of a product integral inequalities Lebesgue–Stieltjes integration by parts |
| title | An Extension of the 1-Dim Lebesgue Integral of a Product of Two Functions |
| title_full | An Extension of the 1-Dim Lebesgue Integral of a Product of Two Functions |
| title_fullStr | An Extension of the 1-Dim Lebesgue Integral of a Product of Two Functions |
| title_full_unstemmed | An Extension of the 1-Dim Lebesgue Integral of a Product of Two Functions |
| title_short | An Extension of the 1-Dim Lebesgue Integral of a Product of Two Functions |
| title_sort | extension of the 1 dim lebesgue integral of a product of two functions |
| topic | extension of the 1-dim Lebesgue integral of a product integral inequalities Lebesgue–Stieltjes integration by parts |
| url | https://www.mdpi.com/2227-7390/11/15/3341 |
| work_keys_str_mv | AT claracarlota anextensionofthe1dimlebesgueintegralofaproductoftwofunctions AT antonioornelas anextensionofthe1dimlebesgueintegralofaproductoftwofunctions AT claracarlota extensionofthe1dimlebesgueintegralofaproductoftwofunctions AT antonioornelas extensionofthe1dimlebesgueintegralofaproductoftwofunctions |