Abstract
We introduce the notion of a measurable function needed in the construction of the integral, study the properties of measurable functions and different types of convergence for sequences of measurable functions. We prove the important theorem of Egorov, which reduces pointwise convergence to uniform convergence by deleting an appropriate set of arbitrarily small measure. We discuss the question of approximation of measurable functions by continuous ones.
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Notes
- 1.
Francesco Paolo Cantelli (1875–1966)—Italian mathematician.
- 2.
Frigyes Riesz (1880–1956)—Hungarian mathematician.
- 3.
Dmitri Fyodorovich Egorov (1869–1931)—Russian mathematician.
- 4.
Maurice René Fréchet (1878–1973)—French mathematician.
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© 2013 Springer-Verlag London
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Makarov, B., Podkorytov, A. (2013). Measurable Functions. In: Real Analysis: Measures, Integrals and Applications. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-5122-7_3
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DOI: https://doi.org/10.1007/978-1-4471-5122-7_3
Publisher Name: Springer, London
Print ISBN: 978-1-4471-5121-0
Online ISBN: 978-1-4471-5122-7
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