Abstract
Based on the notions of measure spaces and measurable maps, we introduce the integral of a measurable map with respect to a general measure. This generalizes the Lebesgue integral that can be found in textbooks on calculus. Furthermore, the integral is a cornerstone in a systematic theory of probability that allows for the definition and investigation of expected values and higher moments of random variables.
In this chapter, we define the integral by an approximation scheme with simple functions. Then we deduce basic statements such as Fatou’s lemma. Other important convergence theorems for integrals follow in Chaps. 6 and 7.
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References
W. Rudin, Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics, 3rd edn. (McGraw-Hill, New York, 1976)
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Klenke, A. (2020). The Integral. In: Probability Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-030-56402-5_4
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DOI: https://doi.org/10.1007/978-3-030-56402-5_4
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