Geodesic
The $M_\alpha $ and $C$-integrals
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 4, pp. 869-878
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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.
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.
DOI : 10.1007/s10587-012-0070-1
Classification : 26A39
Keywords: $M_\alpha $-integral; $ACG_\alpha $ function 
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Park, Jae Myung; Ryu, Hyung Won; Lee, Hoe Kyoung; Lee, Deuk Ho. The $M_\alpha $ and $C$-integrals. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 4, pp. 869-878. doi: 10.1007/s10587-012-0070-1

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