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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
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.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag London
About this chapter
Cite this chapter
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
Download citation
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
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)