Geodesic
An approach to the Lebesgue integral by means of outer integral norm
Trudy Instituta matematiki, Tome 20 (2012) no. 1, pp. 22-32.

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Here we propose a new definition for the Lebesgue integral, based on an outer integral norm. It provides an easy way to prove the standard Levi, Fatou, Lebesgue, and Riesz theorems. 
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V. I. Bakhtin. An approach to the Lebesgue integral by means of outer integral norm. Trudy Instituta matematiki, Tome 20 (2012) no. 1, pp. 22-32. http://geodesic.mathdoc.fr/item/TIMB_2012_20_1_a2/

[1] Riss F., Sekefalvi-Nad B., Lektsii po funktsionalnomu analizu, Mir, M., 1979 | MR

[2] Daniell P. J., “A general form of integral”, Ann. of Math., 19 (1917–1918), 279–294 | DOI | MR