Geodesic
On the interrelation of $A$- and $B$-integrals
Sbornik. Mathematics, Tome 57 (1987) no. 2, pp. 421-435

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Using the law of large numbers, the author shows that on the set of measurable functions the $A$-integral is an extension of the $B$-integral. Also, by using a form of the law of large numbers, he defines an integral that includes the Lebesgue integral and has, for any given function whose positive or negative part is not Lebesgue integrable, any preassigned number as its integral over (0,1). Bibliography: 12 titles. 
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     title = {On the interrelation of $A$- and $B$-integrals},
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B. V. Pannikov. On the interrelation of $A$- and $B$-integrals. Sbornik. Mathematics, Tome 57 (1987) no. 2, pp. 421-435. http://geodesic.mathdoc.fr/item/SM_1987_57_2_a6/