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- W4382808095 abstract "This chapter presents a formal introduction to measure theory. We cover several related, but different, problems. The first problem is the extension of the notion of integration of real-valued functions from the real line to a larger class of spaces. This problem will be addressed by the theory of measurable spaces and Lebesgue integrals. In particular, the theory of Lebesgue integrals will be used to introduce the notion of multiple integrals, that is, the integral of functions of many variables, in a very natural way. There is, however, a price to pay: we have first to build and discuss the Lebesgue measure on the real numbers. This is a rather tedious exercise, but as a final prize, in addition to the possibility of extending the theory of integration to higher-dimensional spaces, one also gains a better understanding of the Riemann integral and its limitations. The material in this chapter is also important for a more formal study of probability theory, statistics, and stochastic processes. A satisfactory coverage of this topic is outside the scope of the present book, but I will introduce basic notions and useful notations. The discussion of the different topics includes various examples and counterexamples aimed at clarifying the results and dispelling myths. The examples will also serve the purpose of adding some intuition to the rather abstract results of formal measure theory." @default.
- W4382808095 created "2023-07-02" @default.
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- W4382808095 date "2023-01-01" @default.
- W4382808095 modified "2023-09-23" @default.
- W4382808095 title "Measure Theory" @default.
- W4382808095 doi "https://doi.org/10.1007/978-3-031-30316-6_9" @default.
- W4382808095 hasPublicationYear "2023" @default.
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