Summary
4.1 This is a short review of base-b expansions of a real number. These expansions will be often used throughout the text and the exercises.
4.2 We now define the famous Cantor singular function: continuous, increasing, but its derivative is 0 almost everywhere (for Lebesgue measure).
4.3 We prove the existence of “pathological” sets. In particular, using Cantor singular function we prove the existence of a non-Borelian negligible set.
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© 1996 Springer-Verlag New York, Inc.
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Simonnet, M. (1996). Lebesgue Measure on R. In: Measures and Probabilities. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4012-9_4
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DOI: https://doi.org/10.1007/978-1-4612-4012-9_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-94644-3
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