Real and Complex Analysis (Rudin)

NOTES AND COMMENTS 403
CbapterlO
See S. N. Mergelyan, Uniform Approximations to Functions of a Complex Variable, Uspehi Mat.
Nauk (N.S.) 7, no. 2 (48), 31-122, 1952; Amer. Math. Soc. Translation No. 101, 1954. Our Theorem
20.5 is Theorem 1.4 in Mergelyan's paper.
A functional analysis proof, based on measure-theoretic considerations, has recently been
published by L. Carleson in Math. Scandinavica, vol. 15, pp. 167-175, 1964.
Appendix
The maximality theorem was first stated by Hausdorff on p. 140 of his book "Grundziige der
Mengenlehre," 1914. The proof of the text is patterned after Sec. 16 of Halmos's book [8]. The
idea to choose g so that g(A) - A has at most one element appears there, as does the term "tower."
The proof is similar to one of Zermelo's proofs of the well-ordering theorem; see Math. Ann., vol. 65,
pp. 107-128, 1908.