Previous |  Up |  Next

# Article

Full entry | PDF   (0.2 MB)
Keywords:
$M_\alpha$-integral; $ACG_\alpha$ function
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.
References:
[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. DOI 10.1112/S0024610700008905 | MR 1771855 | Zbl 0980.26006
[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
[3] Piazza, L. Di: A Riemann-type minimal integral for the classical problem of primitives. Rend. Istit. Mat. Univ. Trieste 34 (2002), 143-153. MR 2013947 | Zbl 1047.26005
[4] Gordon, R. A.: The Integrals of Lebegue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics 4 American Mathematical Society (1994). DOI 10.1090/gsm/004/09 | MR 1288751

Partner of