Geodesic
Integration of Non-Measurable Functions
Canadian mathematical bulletin, Tome 9 (1966) no. 3, pp. 307-330

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This is primarily an expository paper based on (and generalizing) some ideas of J. Pierpont [6], W.H. Young [8], R. L. Jeffery [4] and S. C. Fan [2]. Our aim is to give a simple and easily applicable theory of integration for arbitrary extended - real functions over arbitrary sets in a measure space. This will be achieved by using a generalized version of Pierpont1 s upper and lower integrals (with the upper integral playing the main role), and by appropriately defining the operations in the extended real number system, henceforth denoted by E*, so as to make it a commutative semigroup under addition and multiplication. 
Zakon, Elias. Integration of Non-Measurable Functions. Canadian mathematical bulletin, Tome 9 (1966) no. 3, pp. 307-330. doi: 10.4153/CMB-1966-040-5
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     author = {Zakon, Elias},
     title = {Integration of {Non-Measurable} {Functions}},
     journal = {Canadian mathematical bulletin},
     pages = {307--330},
     year = {1966},
     volume = {9},
     number = {3},
     doi = {10.4153/CMB-1966-040-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1966-040-5/}
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