Real and Complex Analysis (Rudin)

NOTES AND COMMENTS 399
Chapter 6
Sec. 6.3. The constant lin is best possible. See R. P. Kaufman and N. W. Rickert, Bull. Am. Math.
Soc., vol. 72, pp. 672-676, 1966, and (for a simpler treatment) W. W. Bledsoe, Am. Math. Monthly, vol.
77, pp. 180-182, 1970.
Sec. 6.10. von Neumann's proof is in a section on measure theory in his paper: On Rings of
Operators, III, Ann. Math., vol. 41, pp. 94-161,1940. See pp. 124-130.
Sec. 6.15. The phenomenon L'" '" (i!)* is discussed by J. T. Schwartz in Proc. Am. Math. Soc.,
vol. 2, pp. 270-275, 1951, and by H. W. Ellis and D. O. Snow in Can. Math. Bull., vol. 6, pp. 211-229,
1963. See also [7], p. 131, and [28], p. 36.
Sec. 6.19. The references to Theorem 2.14 apply here as well.
Exercise 6. See [17], p. 43.
Exercise 10(g). See [36], vol. I, p. 167.
Chapter 7
Sec. 7.3. This simple covering lemma seems to appear for the first time in a paper by Wiener on the
ergodic theorem (Duke Math. J., vol. 5, pp. 1-18, 1939). Covering lemmas playa central role in the
theory of differentiation. See [SO], [53], and, for a very detailed treatment, [41].
Sec. 7.4. Maximal functions were first introduced by Hardy and Littlewood, in Acta Math., vol.
54, pp. 81-116, 1930. That paper contains proofs of Theorems 8.18, 11.25(b), and 17.11.
Sec. 7.21. The same conclusion can be obtained under somewhat weaker hypotheses; see [16],
Theorems 260-264. Note that the proof of Theorem 7.21 uses the existence and integrability of only
the right-hand derivative of f, plus the continuity of f. For a further refinement, see P. L. Walker,
Amer. Math. Monthly, vol. 84, pp. 287-288, 1977.
Secs. 7.25, 7.26. This treatment of the change of variables formula is quite similar to D.
Varberg's in Amer. Math. Monthly, vol. 78, pp. 42-45, 1971.
Exercise 5. A very simple proof, due to K. Stromberg, is in Proc. Amer. Math. Soc., vol. 36,
p.308,1972.
Exercise 12. For an elementary proof that every monotone function (hence every function of
bounded variation) is differentiable a.e., see [24], pp. 5-9. In that work, this theorem is made the
starting point of the Lebesgue theory. Another, even simpler, proof by D. Austin is in Proc. Amer.
Math. Soc., vol. 16, pp. 220-221, 1965.
Exercise 18. These functions CPo are the so-called Rademacher functions. Chap. V of [36] contains
further theorems about them.
ChapterS
Fubini's theorem is developed here as in [7] and [28]. For a different approach, see [25]. Sec. 8.9(c) is
in Fundamenta Math., vol. 1, p. 145,1920.
Sec. 8.18. This proof of the Hardy-Littlewood theorem (see the reference to Sec. 7.4) is essentially
that of a very special case of the Marcinkiewicz interpolation theorem. A full discussion of the
latter may be found in Chap. XII of [36]. See also [50].
Exercise 2. Corresponding to the idea that an integral is an area under a curve, the theory of the
Lebesgue integral can be developed in terms of measures of ordinate sets. This is done in [16].
Exercise 8. Part (b), in even more precise form, was proved by Lebesgue in J. Mathematiques,
ser. 6, vol. 1, p. 201, 1905, and seems to have been forgotten. It is quite remarkable in view of another
example of Sierpinski (Fundamenta Math., vol. 1, p. 114, 1920): There is a plane set E which is not
Lebesgue measurable and which has at most two points on each straight line. Iff= lE' thenfis not
Lebesgue measurable, although all of the sections fx and p are upper semicontinuous; in fact, each
has at most two points of discontinuity. (This example depends on the axiom of choice, but not on the
continuum hypothesis.)