Upper and lower integral difference functionals, closest approximations, and integrability
Rendiconti del Seminario Matematico della Università di Padova, Tome 52 (1974), pp. 1-24. 
@article{RSMUP_1974__52__1_0,
     author = {Appling, William D. L.},
     title = {Upper and lower integral difference functionals, closest approximations, and integrability},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {1--24},
     publisher = {Seminario Matematico of the University of Padua},
     volume = {52},
     year = {1974},
     mrnumber = {404565},
     zbl = {0325.28006},
     language = {en},
     url = {http://www.numdam.org/item/RSMUP_1974__52__1_0/}
}
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Appling, William D. L. Upper and lower integral difference functionals, closest approximations, and integrability. Rendiconti del Seminario Matematico della Università di Padova, Tome 52 (1974), pp. 1-24. http://www.numdam.org/item/RSMUP_1974__52__1_0/

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[2] W.D.L. Appling, Integral representations and refinement unboundedness, Proc. Amer. Math. Soc., 19 (1968), 837-842. | MR | Zbl

[3] W.D.L. Appling, Summability of real-valued set functions, Riv. Mat. Parma, (2), 8 (1967), 77-100. | MR | Zbl

[4] W.D.L. Appling, Continuity and set function summability, Annali di Matematica pura e applicata, 87 (1970), 357-374. | MR | Zbl

[5] W.D.L. Appling, A generalization of absolute continuity and of an analogue of the Lebesgue decomposition theorem, Riv. Mat. Parma, accepted for publication; to appear. | MR | Zbl

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[7] A. Kolmogoroff, Untersuchungen uber den Integralbegriff, Math. Ann., 103 (1930), 654-696. | JFM | MR