Abstract
There are two basic avenues to integration. In the modern approach the integral is introduced first for simple functions—as a weighted average of the values of the function—and then defined for any nonnegative measurable Function f as a limit of the integrals of simple nonnegative functions increasing to f. Conceptually this is extremely simple, but a certain price is paid in terms of proofs. The alternative classical approach, while employing a less intuitive definition, achieves considerable simplicity in proofs of elementary properties.
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Chow, Y.S., Teicher, H. (1997). Integration in a Probability Space. In: Probability Theory. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1950-7_4
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DOI: https://doi.org/10.1007/978-1-4612-1950-7_4
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