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Upper integral and its geometric meaning. (English) Zbl 1164.28003
The main idea of the paper is to study Hahn’s definition of the integral for non-measurable functions. The equivalence to the Fan integral is proved as well as Minkowski type inequality and Chebyshev inequality.
MSC:
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
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References:
[1] FAN S. C.: Integration with respect to an upper-measure function. Amer. J. Math. 63 (1941), 319-337. · Zbl 0025.03401
[2] HAHN H.-ROSENTHAL A.: Set Functions. University of New Mexico Press, Albuquerque, 1948. · Zbl 0033.05301
[3] ROYDEN H. L.: Real Analysis. (3rd, Macmillan Publishing Company, New York, 1988. · Zbl 0704.26006
[4] TAYLOR A. E.: General Theory of Functions and Integration. Dover Publications Inc., New York, 1985.
[5] WARD A. J.: The Fan integrals interpreted as measures in a product-space. Amer. J. Math. 66 (1944), 144-160. · Zbl 0063.08175
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