The $M_{\alpha }$ and $C$-integrals

Park, Jae Myung, et al. "The $M_\alpha $ and $C$-integrals." Czechoslovak Mathematical Journal 62.4 (2012): 869-878. <http://eudml.org/doc/246511>.

@article{Park2012,
abstract = {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 $F^\{\prime \}=f$ 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.},
author = {Park, Jae Myung, Ryu, Hyung Won, Lee, Hoe Kyoung, Lee, Deuk Ho},
journal = {Czechoslovak Mathematical Journal},
keywords = {$M_\alpha $-integral; $ACG_\alpha $ function; -integral; -integral; McShane integral; Henstock-Kurzweil integral},
language = {eng},
number = {4},
pages = {869-878},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The $M_\alpha $ and $C$-integrals},
url = {http://eudml.org/doc/246511},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Park, Jae Myung
AU - Ryu, Hyung Won
AU - Lee, Hoe Kyoung
AU - Lee, Deuk Ho
TI - The $M_\alpha $ and $C$-integrals
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 4
SP - 869
EP - 878
AB - 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 $F^{\prime }=f$ 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.
LA - eng
KW - $M_\alpha $-integral; $ACG_\alpha $ function; -integral; -integral; McShane integral; Henstock-Kurzweil integral
UR - http://eudml.org/doc/246511
ER -