Measure Theory — Basic Notions
Measurable sets and functions; measures and integration; mono tone and dominated convergence; transformation of integrals; product measures and Fubini’s theorem; Lp-spaces and projec tion; approximation; measure spaces and kernels.
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- The first author to consider measures in the modern sense was Borel (1895, 1898), who constructed Lebesgue measure on the Borel σ-field in R. The corresponding integral was introduced by Lebesgue (1902, 1904), who also established the dominated convergence theorem. The monotone convergence theorem and Fatou’s lemma were later obtained by Levi (1906a) and Fatou (1906), respectively. Lebesgue also introduced the higher-dimensional Lebesgue measure and proved a first version of Fubini’s theorem, subsequently generalized by Fubini (1907) and Tonelli (1909). The integration theory was extended to general measures and abstract spaces by many authors, including Radon (1913) and Frechet (1928).Google Scholar
- The norm inequalities in Lemma 1.29 were first noted for finite sums by Hölder (1889) and Minkowski (1907), respectively, and were later extended to integrals by Riesz (1910). Part (i) for p = 2 goes back to Cauchy (1821) for finite sums and to Buniakowsky (1859) for integrals. The Hilbert space projection theorem can be traced back to Levi (1906b).Google Scholar
- The monotone class Theorem 1.1 was first proved, along with related results, already by Sierpiski (1928), but the result was not used in probability theory until Dynkin (1961). More primitive versions had previously been employed by Halmos (1950) and Doob (1953).Google Scholar
- Most results in this chapter are well known and can be found in any textbook on real analysis. Many probability texts, including Loève (1977) and Billingsley (1995), contain detailed introductions to measure theory. There are also some excellent texts in real analysis adapted to the needs of probabilists, such as Dudley (1989) and Doob (1994). The former author also provides some more detailed historical information.Google Scholar