Notes on integration I: The underlying convergence theorem

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, VOL. x, 357-360 (1957)
<strong>Notes</strong> on Integration I:
The Underlying Convergence Theorem *
W. F. EBERLEIN**
University of Wisconsin and New York University
1. Introduction
Let C denote the (algebra and lattice of) real-valued continuous func-
tions on a compact Hausdorff space S. An integral over S is a positive linear
functional on C-that is, a mapping I of C into the real numbers satisfying
(1) I(af+bg) = al(f)+bl(g),
(2) I(/) 2 0 whenever f 2 0.
The following proposition dates back to Arzelal, 1885, [l] and Osgood,
1897, [8] in the classical case: S = the closed finite interval (a, b) andI(f) =
J: f ( WX.
THEOREM. Let limn f, = f fiointwise (f,, f in C) and If,l 5 M < co for
all n. Then lim,I(f,) = I(/).
This result plays a paradoxical role in integration theory? Although
the theorem concerns only continuous functions on S, the machinery of con-
ventional proofs stemming from Borel-Lebesgue (c. 1900, measure theory
[7]) or Daniel1 (1917, functional extension [4]) involves a preliminary ex-
tension of the mapping I to a larger class of (measurable) functions on S.
On the other hand it is known (cf. Banach, 1937, [3] and F. Riesz, 1952, [9])
that the theorem itself is a natural basis for extending I. This all but circu-
lar state of affairs may account for the variety of “elementary” proofs-
of varying degrees of generality-devised by such mathematicians as
Riesz (1917, [9] and 1952, [lo]) and Hausdorff (1927, 1151) (disguised ex-
tension theory), Bieberbach-Landau (1918, set of measure 0 theory [GI),
Banach (1932, set theory [2]) and others.
*This paper represents results obtained under the sponsorship of the National Science
Foundation, Contracts NSF-G2052 and NSF-G3050, and the Air Office of Scientific Research,
Contract AF lS(600)-1634.
**The author is a Temporary Member of the Institute of Mathematical Sciences,
New York University, for the academic year 1956-57.
’Arzela’s paper actually deals with sequences of Riemann integrable functions.
*For a discussion of the role of the theorem in modern functional analysis see Wymore,
1955, [12].
357

358 W. F. EBERLEIN
Here is a new and elementary proof of the general theorem-the basis
of a neogeometric approach to integration over function spaces, with ap-
plications to numerical analysis and physics. The author is indebted to
T. H. Hildebrandt and P. D. Lax for bibliographical criticism.
2. Proof of the Theorem
We show first that compactness of S makes I "countably positive":
If\ 5 2';"
LEMMA 1. (stones)
\/,I implies r(Ifl) 5 CTI(lf,l).
Proof: Given E > 0 and z in S, there exists an integer N(x) such that
[ f (z) I < c:@) If, (z) 1 + E, and the inequality persists in some neighborhood U (x)
of 2 because of continuity. Since S is compact, the open covering (U(z))
(x in S) thus obtained contains a finite subcovering {U(xj)). Let N be the
largest iV(zi). Then If1 2 2: If,/+& implies
N
Wl) r 2 J(ILl)+EW
1
co
Z'(lf~l)+E'(l)J
1
whence I(lf1) 5 CyI(lf,l), since > 0 was arbitrary.
We require some elementary properties of the L p norms on C in the
distinguished cases p = 1, 2, co. Set llfllm = sup If(x)l (x in S) and note
that (1) and (2) imply
(A) II(f)l 5 J(lfl) 5 Ilfllm * I(1).
Thus we may first assume I( 1) > 0 (non-trivial case) and then (on replacing
I by a suitable constant multiple)
(3) I(1) = 1.
Next set (1, g) = I(fg) (f, g in C) and = (1, f)% = Finally set
IlflII = I(lfl). Then:
(B) I (f, g) I 5 I If1 12 * I Igl I2 (Schwarz inequality) 3
(C) Ilf+slI~+fIf-gll; = 2{1lfll~+llg11;~ (parallelogram identity),
(Dl Ilflll -= I(If1) = (If19 1) 5 llf112 ' lllll2 = Ilfll2>
(E) I If1 1; = I(f7 5 I lf21 Im = I If1 1: *
3This proposition is actually the basic axiom in Stone's theory of integration (1948,
[I l]), while that of the theorem is essentially the basic axiom in Banach's theory (1937, [2]).

NOTES ON INTEGRATION I 359
Consider now some preliminary reductions of the theorem itself. We
may assume f = 0 and f, 1 0 for all n (otherwise replace f, by If,-fl).
Clearly L = lim sup, I(/,) 5 M exists, and we need only show L = 0. But
we may assume limn I(f,) = L (otherwise pass to a subsequence). The pro-
position then becomes the
REDUCED THEOREM. Let: (a) f,ZieinCfor aZZn, (b) supnIlfnlIm SM

360 W. F. EBERLEIN
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[I21 Wymore, A. W., On Weak Compactness in Functional Analysis, Thesis, University of
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Received January, 1957.