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Integrals on Measure Spaces

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Lectures on Functional Analysis and the Lebesgue Integral

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Abstract

In Chap. 5 we defined the Lebesgue integral of functions defined on \(\mathbb{R}\). In this chapter we show that the theory remains valid in a much more general framework;moreover, almost all proofs can be repeated word for word. The results of this chapter include integrals of several variables and integrals on probability spaces

In my opinion, a mathematician, in so far as he is a mathematician, need not preoccupy himself with philosophy – an opinion, moreover, which has been expressed by many philosophers. –H. Lebesgue

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Notes

  1. 1.

    Radon [366], Fréchet [158], Daniell [93]. In this book we consider only real-valued functions, although Bochner [46] extended the theory to Banach space-valued functions, and this has important applications among others in the theory of partial differential equations. See, e.g., Dunford–Schwartz [117], Edwards [119], Yosida [488], and Lions–Magenes [305].

  2. 2.

    Kolmogorov [252].

  3. 3.

    Halmos [184] introduced a slightly more restricted notion, but the present definition has become standard by now.

  4. 4.

    Borel [59].

  5. 5.

    Since \(\mu (\varnothing ) = 0\), the equality (7.1) holds for finite disjoint sequences as well. Finitely additive set functions were studied before Borel by Harnack [192], Cantor [74, pp. 229–236], Stolz [437], Peano [353, pp. 154–158] and Jordan [231, pp. 76–79].

  6. 6.

    The statement and its proof remain valid for unbounded intervals, too.

  7. 7.

    In this book we do not distinguish between different infinite cardinalities, except in an example on p. 243 and in some exercises.

  8. 8.

    The so-called co-finite sets.

  9. 9.

    The measurability of the constant functions is equivalent to the measurability of X.

  10. 10.

    In this book, following F. Riesz, we adopt a more restrictive measurability notion than usual. See Sect. 7.7 on the advantages of this choice.

  11. 11.

    We apply the lemma for each f n , and we take the union of the corresponding set sequences.

  12. 12.

    We sometimes express this property by saying that f has a \(\sigma\)-finite support . Using this terminology X is measurable \(\Longleftrightarrow\ X\) is \(\sigma\)-finite.

  13. 13.

    The function class C 1 was defined on p. 174.

  14. 14.

    We already know that this extension is unique.

  15. 15.

    See the beginning of the proof of Proposition 7.6: we already know that \(\mathcal{P}\subset \mathcal{N} \subset \mathcal{M}\).

  16. 16.

    Euler [130], Dirichlet [113], and Stolz [438, pp. 93–94].

  17. 17.

    Lebesgue [288] (for bounded functions), Fubini [164]. Fubini’s proof was incorrect; the first correct proofs were given by Hobson [214] and de la Vallée-Poussin [464]. See Hawkins [198].

  18. 18.

    In the proof of this theorem the application of Lemma 5.3 (p. 173) is sufficient because we consider only sequences of step functions.

  19. 19.

    Tonelli [457].

  20. 20.

    The former counterexamples of Cauchy [81, p. 394], Thomae [452] and du Bois-Reymond [53] were based on the smallness of the class of Riemann integrable functions.

  21. 21.

    Further counterexamples are given in Exercise 7.8 below, p. 253.

  22. 22.

    See Gurevich–Silov [175, p. 180].

  23. 23.

    For otherwise we would have for every one-point set \(A \subset N\) the impossible inequalities \(1 =\mu (A) =\mu (A \cap N) \leq 0\).

  24. 24.

    This also follows from Lemma 7.13 (c) below.

  25. 25.

    We may have equality if \(\mu (A) = -\infty \).

  26. 26.

    We recall that they are defined on a \(\sigma\)-ring.

  27. 27.

    See Proposition 7.11, p. 230.

  28. 28.

    See Proposition 7.3, p. 216.

  29. 29.

    Lebesgue [295, pp. 232–249].

  30. 30.

    See p. 213.

  31. 31.

    However, the present definition of absolute continuity is interesting only if μ is a measure.

  32. 32.

    Radon [366, pp. 1342–1351] and Nikodým [342, pp. 167–179]. We recall from the preceding section that the strong \(\sigma\)-finiteness condition is satisfied if μ is a finite measure or if X is measurable.

  33. 33.

    See also an alternative proof of von Neumann [339, pp. 124–130], based on the orthogonal projection in Hilbert spaces.

  34. 34.

    See the remark on p. 238

  35. 35.

    See Halmos [184, pp. 131–132]. In this example we use the notion of cardinality of infinite sets, but we need only the simplest results: see, e.g., Halmos [186] or Vilenkin [467, 468].

  36. 36.

    Because every measurable set is covered by countably many lines.

  37. 37.

    See Proposition 5.17 (d), p. 190.

  38. 38.

    See Hewitt–Stromberg [207, p. 317].

  39. 39.

    See Lemma 7.13 (b), p. 232.

  40. 40.

    The proposition extends classical results of Euler [130, p. 303], Lagrange [280, p. 624] and Jacobi [224, p. 436].

  41. 41.

    They are also ν-measurable because \(\mu (X) < \infty \).

  42. 42.

    Proposition 5.17 (e), p. 190.

  43. 43.

    We recall from Lemma 7.5 (p. 220) that X is measurable \(\Longleftrightarrow\) it has a countable cover by sets of \(\mathcal{P}\) (and hence of finite measure).

  44. 44.

    Indeed, this choice is taken by most contemporary textbooks by defining measurability using inverse images. While Hausdorff’s elegant characterization of continuous functions by inverse images of open or closed sets is extremely useful in topology, the analogous definition of measurability leads to several annoying counterexamples.

  45. 45.

    Fréchet [158] and Riesz–Sz.-Nagy [394].

  46. 46.

    L. Czách, private communication, 2005.

  47. 47.

    Carathéodory [77]. See also Burkill [68], Halmos [184], and Natanson [332].

  48. 48.

    Stieltjes [435].

  49. 49.

    Hausdorff [196]. See, e.g., Falconer [134]. Some number-theoretical applications are given in de Vries–Komornik [101] and Komornik–Kong–Li [259].

  50. 50.

    More generally, we may consider countable covers by sets of diameter \(\mathop{\mathrm{diam}}\nolimits I_{i} \leq \delta\) in a metric space.

  51. 51.

    Carathéodory’s construction (Exercise 7.4) yields the s-dimensional Hausdorff measure.

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    Vilmos Komornik

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Komornik, V. (2016). Integrals on Measure Spaces. In: Lectures on Functional Analysis and the Lebesgue Integral. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-6811-9_7

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