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An Integral Over Function Space

Published online by Cambridge University Press:  20 November 2018

W. F. Eberlein
W. F. Eberlein*
Affiliation:
University of Rochester
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Real functions

may be identified with elements x = (x0, x1, x2, … ) of the sequence space l1 Since the unit sphere S∞ of l1 is compact under the weak* topology = topology of co-ordinatewise convergence, a countably additive measure on S∞ is induced by a positive linear functional E (integral) on C(S∞), the weak* continuous real-valued functions on S∞. There exists a natural integral over S∞ reducing to

when f is a function of x0 alone. The partial sums Sn = Sn(x) of the power series for x(t) then form a martingale and zero-or-one phenomena appear. In particular, if R(x) is the radius of convergence of the series and e is the base of the natural logarithms, it turns out that R(x) = e for almost all x in S∞. Applications of the integral to the theory of numerical integration, the original motivation, will appear in a later paper.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 379 - 384
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Banach, S., The Lebesgue integral in abstract spaces, note to S. Saks, Theory of the integral (Warsaw, 1933).Google Scholar
2. Doob, J. L., Stochastic processes (New York, 1953).Google Scholar
3. Dunford, N. and Schwartz, J. T., Linear operators (New York, 1958).Google Scholar