Measure Theory — Key Results
Outer measures and extension; Lebesgue and Lebesgue-Stieltjes measures; Jordan-Hahn and Lebesgue decompositions; Radon Nikodým theorem; Lebesgue’s differentiation theorem; functions of finite variation; Riesz’ representation theorem; Haar and invariant measures.
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- As we have seen, Borel (1995, 1998) was the first to prove the existence of one-dimensional Lebesgue measure. However, the modern construction via outer measures in due to Caratheodory (1918).Google Scholar
- Functions of bounded variation were introduced by Jordan (1881), who proved that any such function is the difference of two nondecreasing functions. The corresponding decomposition of signed measures was obtained by Hahn (1921). Integrals with respect to nondecreasing functions were defined by Stieltjes (1894), but their importance was not recognized until Riesz (1909b) proved his representation theorem for linear functionals on C[0,1]. The a.e. differentiability of a function of bounded variation was first proved by Lebesgue (1904).Google Scholar
- Vitali (1905) was the first author to see the connection between absolute continuity and the existence of a density. The Radon-Nikodym theorem was then proved in increasing generality by Radon (1913), Daniell (1920), and Nikodym (1930). The idea of a combined proof that also establishes the Lebesgue decomposition is due to von Neumann.Google Scholar
- Invariant measures on specific groups were early identified through explicit computation by many authors, notably by Hurwitz (1897) for the case of SO(n). Haar (1933) proved the existence (but not the uniqueness) of invariant measures on an arbitrary lcscH group. The modern treatment originated with Weil (1940), and excellent expositions can be found in many books on real or harmonic analysis. Invariant measures on more general spaces are usually approached via quotient spaces. Our discussion in Theorem 2.29 is adapted from Royden (1988).Google Scholar