| Title:
|
The $M_\alpha $ and $C$-integrals (English) |
| Author:
|
Park, Jae Myung |
| Author:
|
Ryu, Hyung Won |
| Author:
|
Lee, Hoe Kyoung |
| Author:
|
Lee, Deuk Ho |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
4 |
| Year:
|
2012 |
| Pages:
|
869-878 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we define the $M_\alpha $-integral of real-valued functions defined on an interval $[a,b]$ and investigate important properties of the $M_{\alpha }$-integral. In particular, we show that a function $f\colon [a,b]\rightarrow R$ is $M_{\alpha }$-integrable on $[a,b]$ if and only if there exists an $ACG_{\alpha }$ function $F$ such that $F'=f$ almost everywhere on $[a,b]$. It can be seen easily that every McShane integrable function on $[a,b]$ is $M_{\alpha }$-integrable and every $M_{\alpha }$-integrable function on $[a,b]$ is Henstock integrable. In addition, we show that the $M_{\alpha }$-integral is equivalent to the $C$-integral. (English) |
| Keyword:
|
$M_\alpha $-integral |
| Keyword:
|
$ACG_\alpha $ function |
| MSC:
|
26A39 |
| idZBL:
|
Zbl 1274.26016 |
| idMR:
|
MR3010244 |
| DOI:
|
10.1007/s10587-012-0070-1 |
| . |
| Date available:
|
2012-11-10T21:25:01Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143031 |
| . |
| Reference:
|
[1] Bongiorno, B., Piazza, L. Di, Preiss, D.: A constructive minimal integral which includes Lebesgue integrable functions and derivatives.J. Lond. Math. Soc., II. Ser. 62 (2000), 117-126. Zbl 0980.26006, MR 1771855, 10.1112/S0024610700008905 |
| Reference:
|
[2] Bruckner, A. M., Fleissner, R. J., Fordan, J.: The minimal integral which includes Lebesgue integrable functions and derivatives.Colloq. Math. 50 (1986), 289-293. MR 0857865, 10.4064/cm-50-2-289-293 |
| Reference:
|
[3] Piazza, L. Di: A Riemann-type minimal integral for the classical problem of primitives.Rend. Istit. Mat. Univ. Trieste 34 (2002), 143-153. Zbl 1047.26005, MR 2013947 |
| Reference:
|
[4] Gordon, R. A.: The Integrals of Lebegue, Denjoy, Perron, and Henstock.Graduate Studies in Mathematics 4 American Mathematical Society (1994). MR 1288751, 10.1090/gsm/004/09 |
| . |
