## Document Type

Article

## Department

Mathematics (HMC)

## Publication Date

1964

## Abstract

Let X = {x1, x2, ...} be a countably infinite topological space; then the space C*(X) of all bounded real-valued continuous functions f may be regarded as a space of sequences (f(x1), f(x2), ...). It is well known [7, p. 54] that no regular (Toeplitz) matrix can sum all bounded sequences. On the other hand, if (x1, x2, ...) converges in X (to x_{m}), then every regular matrix sums all f in C*(X) (to f(x_{m})).

The main result of this paper is that if a regular matrix sums all f in C*(X) then it sums f to Σαnf(xn), for some absolutely convergent series Σαn. We use this to show that no regular matrix can sum all of C*(X) if X is extremally disconnected (the closure of every open set is open). This extends a theorem of W. Rudin [6], which has an equivalent hypothesis (X is embeddable in the Stone-Cech compactification βN of a discrete space) and concludes that not all f in C*(X) are Cesàro summable.

For any continuous linear functional φ on C*(X) one has a ("Riesz") representation φ(f) = ∫fdμ, where μ is a Radon measure on βX. Our main result is just that X supports μ; μ is forced to be atomic since X is countable. We show further that X has a subset T, the set of *heavy points*, such that the functionals we are concerned with correspond exactly to measures μ supported by T with μ(T) = 1. Our knowledge of T is limited; it will be summarized elsewhere.

## Rights Information

© 1964 American Mathematical Society

## Terms of Use & License Information

## DOI

10.1090/S0002-9904-1964-11127-7

## Recommended Citation

Henriksen, Melvin, J.R. Isbell. "Averages of continuous functions on countable spaces." Bulletin of the American Mathematical Society 70.6 (1964): 287–290. DOI: 10.1090/S0002-9904-1964-11127-7

## Comments

Previously linked to as: http://ccdl.libraries.claremont.edu/u?/irw,293

This article can also be found at http://www.ams.org/bull/1964-70-02/S0002-9904-1964-11127-7/home.html