Abstract
In this chapter, we lay the measure theoretic foundations of probability theory. We introduce the classes of sets (semirings, rings, algebras, σ-algebras) that allow for a systematic treatment of events and random observations. Using the measure extension theorem, we construct measures, in particular probability measures on σ-algebras. Finally, we define random variables as measurable maps and study the σ-algebras generated by certain maps.
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Notes
- 1.
By “countable” we always mean either finite or countably infinite.
- 2.
Warning: For some authors, the geometric distribution is shifted by one to the right; that is, it is a distribution on \({\mathbb {N}}\).
References
Dudley RM (2002) Real analysis and probability. Cambridge studies in advanced mathematics, vol 74. Cambridge University Press, Cambridge. Revised reprint of the 1989 original
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Klenke, A. (2014). Basic Measure Theory. In: Probability Theory. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-5361-0_1
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DOI: https://doi.org/10.1007/978-1-4471-5361-0_1
Publisher Name: Springer, London
Print ISBN: 978-1-4471-5360-3
Online ISBN: 978-1-4471-5361-0
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