Summability integrals

Author:
George Brauer

Journal:
Proc. Amer. Math. Soc. **61** (1976), 85-89

MSC:
Primary 40C05; Secondary 28A25

DOI:
https://doi.org/10.1090/S0002-9939-1976-0440242-9

MathSciNet review:
0440242

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Abstract: An integral on the set of natural numbers is defined. If is a subset of such that the characteristic function of is evaluated to some number by a regular nonnegative summation matrix , then is defined to be defines a finitely additive measure on . If is a sequence which can be written as a linear combination of characteristic functions , where each sequence is evaluated by , then is defined to be . Finally the definition of the integral is naturally extended to , the class of sequences which can be approximated by linear combinations of characteristic functions [2, pp. 85-88]. It is shown that if and are two nonnegative regular matrices such that the convergence field of includes that of , then includes provided is normal. Finally for a nonnegative regular matrix , the spaces spanned by sequences such that is bounded and exists are studied. It is shown that if is greater than one, then the sequences in give rise to a set of bounded linear functionals on which are weak star dense in the dual of .

**[1]**J. R. Edwards and S. G. Wayment,*A summability integral*, J. Reine Angew. Math.**255**(1972), 85–93. MR**0304919****[2]**Leonard Gillman and Meyer Jerison,*Rings of continuous functions*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0116199**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0440242-9

Keywords:
Linear combinations of characteristic functions,
linear functionals,
dual,
weak star dense

Article copyright:
© Copyright 1976
American Mathematical Society