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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Summability integrals


Author: George Brauer
Journal: Proc. Amer. Math. Soc. 61 (1976), 85-89
MSC: Primary 40C05; Secondary 28A25
MathSciNet review: 0440242
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Abstract: An integral on the set of natural numbers $ N$ is defined. If $ E$ is a subset of $ N$ such that the characteristic function $ {\chi _E}$ of $ E$ is evaluated to some number $ \sigma $ by a regular nonnegative summation matrix $ A$, then $ {\mu _A}(E)$ is defined to be $ \sigma ;{\mu _A}$ defines a finitely additive measure on $ N$. If $ s$ is a sequence which can be written as a linear combination of characteristic functions $ \Sigma _{i = 1}^n{a_i}{\chi _{{E_i}}}$, where each sequence $ {\chi _{{E_i}}}$ is evaluated by $ A$, then $ {\smallint _N}sd{\mu _A}$ is defined to be $ \Sigma {a_i}{\mu _A}({E_i})$. Finally the definition of the integral is naturally extended to $ L(A)$, the class of sequences which can be approximated by linear combinations of characteristic functions [2, pp. 85-88]. It is shown that if $ A$ and $ B$ are two nonnegative regular matrices such that the convergence field of $ A$ includes that of $ B$, then $ L(A)$ includes $ L(B)$ provided $ B$ is normal. Finally for a nonnegative regular matrix $ A = ({a_{nk}})$, the spaces $ {L^p}(A)$ spanned by sequences such that $ \{ \Sigma _{k = 0}^\infty {a_{nk}}\vert{s_k}{\vert^p}\} $ is bounded and $ \lim \Sigma _{k = 0}^\infty {a_{nk}}s_k^p$ exists are studied. It is shown that if $ p$ is greater than one, then the sequences in $ {L^{p'}}(A)$ give rise to a set of bounded linear functionals on $ {L^p}(A)$ which are weak star dense in the dual of $ {L^p}(A)$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0440242-9
PII: S 0002-9939(1976)0440242-9
Keywords: Linear combinations of characteristic functions, linear functionals, dual, weak star dense
Article copyright: © Copyright 1976 American Mathematical Society