Abstract
Here we continue on our journey toward defining the Lebesgue integral. The previous chapter developed the theory of positive measures and measurable sets which will play the roles filled by length and intervals in the Riemann theory. The next phase of this process is to determine the class of functions to which we will apply our new integral. The final step will then be to construct the Lebesgue integral. As previously mentioned, this new integral will be concerned more specifically with the range of a function rather than its domain and will therefore be defined for functions which map into \(\mathbb{R}\), but whose domain may be some other set, be it of numbers or other objects. In Sect.7.5 and the introduction to Chap. 8 we spent a considerable amount of time discussing some of the major deficiencies of the Riemann integral. Once we have our formulation for the Lebesgue integral, we will demonstrate that it is a vast improvement upon its predecessor.
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Pons, M.A. (2014). Lebesgue Integration. In: Real Analysis for the Undergraduate. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9638-0_9
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