Geodesic
Henstock-Kurzweil and McShane product integration; descriptive definitions
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 1, pp. 241-269 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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The Henstock-Kurzweil and McShane product integrals generalize the notion of the Riemann product integral. We study properties of the corresponding indefinite integrals (i.e. product integrals considered as functions of the upper bound of integration). It is shown that the indefinite McShane product integral of a matrix-valued function $A$ is absolutely continuous. As a consequence we obtain that the McShane product integral of $A$ over $[a,b]$ exists and is invertible if and only if $A$ is Bochner integrable on $[a,b]$.
The Henstock-Kurzweil and McShane product integrals generalize the notion of the Riemann product integral. We study properties of the corresponding indefinite integrals (i.e. product integrals considered as functions of the upper bound of integration). It is shown that the indefinite McShane product integral of a matrix-valued function $A$ is absolutely continuous. As a consequence we obtain that the McShane product integral of $A$ over $[a,b]$ exists and is invertible if and only if $A$ is Bochner integrable on $[a,b]$.
Classification : 26A39, 28B05, 46G10
Keywords: Henstock-Kurzweil product integral; McShane product integral; Bochner product integral 
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Slavík, Antonín; Schwabik, Štefan. Henstock-Kurzweil and McShane product integration; descriptive definitions. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 1, pp. 241-269. http://geodesic.mathdoc.fr/item/CMJ_2008_58_1_a14/

[1] J. D. Dollard and C. N. Friedman: Product Integration with Applications to Differential Equations. Addison-Wesley Publ. Company, Reading, Massachusetts, 1979. | MR

[2] J. Jarník and J. Kurzweil: A general form of the product integral and linear ordinary differential equations. Czech. Math. J. 37 (1987), 642–659. | MR

[3] P. R. Masani: Multiplicative Riemann integration in normed rings. Trans. Am. Math. Soc. 61 (1947), 147–192. | DOI | MR | Zbl

[4] Š. Schwabik: Bochner product integration. Math. Bohem. 119 (1994), 305–335. | MR | Zbl

[5] Š. Schwabik: The Perron product integral and generalized linear differential equations. Časopis pěst. mat. 115 (1990), 368–404. | MR | Zbl

[6] Š. Schwabik and Ye Guoju: Topics in Banach Space Integration. World Scientific, Singapore, 2005. | MR