Proc. Amer. Math. Soc. 61 (1976), 85-89
Primary 40C05; Secondary 28A25
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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 .
R. Edwards and S.
G. Wayment, A summability integral, J. Reine Angew. Math.
255 (1972), 85–93. MR 0304919
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
- J. R. Edwards and S. G. Wayment, A summability integral, J. Reine Angew Math. 255 (1972), 85-93. MR 46 #4050. MR 0304919 (46:4050)
- L. Gillman and M. Jerison, Rings of continuous functions, Univ. Ser. in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #6994. MR 0116199 (22:6994)
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Linear combinations of characteristic functions,
weak star dense
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