Maurice Heins - Selected Topics in the Classical Theory of Functions of a Complex Variable

Selected Topics in the
Classical Theory of Functions
of a Complex Variable

Maurice Heins

DOVER PUBLICATIONS, INC.

Mineola, New York

Appendix

1. Riesz Representation Theorem for C

In this article of the appendix we shall give a proof of the Riesz representation theorem for positive additive functionals on C:

Given , there exists a nondecreasing function on R satisfying (x + 2) = (x) + (1) such that

The proof given here is applicable to other classes of continuous functions.

PROOF: Given f C, it will be convenient to introduce the norm of f, . For each positive integer n, we introduce first

and then

Now n C. Further,

Given f C, we introduce

and note that

where f is the modulus of continuity of f. [Consider f(x) fn(x) on an interval {2k/nx 2{k + 1)/n}, and observe that fn is linear on this interval and takes the same values as/at the endpoints.] Hence

Further,

We define

so that

We define n elsewhere by the requirement that, for all x, n is to satisfy

We now see that

By the selection principle we are assured of the existence of a subsequence such that for each rational number q,the sequence possesses a finite limit. We define by

The function is a monotone nondecreasing function satisfying

and

We assert that has the desired property:

To see that this is so, we introduce a finite increasing sequence , where t0 = 0, ts = 2 and each tk is a rational multiple of 2. Let = max (tk+1tk), k = 0, s 1. From (8) and the Standard appraisals relating Riemann-Stieltjes integrals and their approximating sums, we have

and

f being the modulus of continuity of f. Hence,

and, consequently,

Similarly,

We conclude that

The theorem is established.

2. Lebesgues Theorem

An account of F. Rieszs proof for the continuous case of Lebesgues theorem is to be found in F. Riesz and B. Sz.-Nagy, Leons d Analyse Fonctionnelle, Budapest, 1952.

As far as I am aware, Riesz never gave a detailed account of the proof for the general case. He States on several occasions (ibid., p. 9) that it calls only for a few simple modifications of the proof for the continuous case. Since we want to use the Lebesgue theorem unrestrictedly and wish to keep the exposition self-contained, we shall in fact consider the general theorem (cf. J. von Neumann, Functional Operators, vol. 1, Princeton, 1950).

The core of Rieszs argument is his celebrated rising-sun lemma. We consider a function g with domain a bounded closed interval {axb}, a < b,