Vol. 34, No. 3, 1970

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Some measure algebras on the integers

Richard A. Scoville

Vol. 34 (1970), No. 3, 769–779

The author constructs some abstract algebras whose elements are subsets of the positive integers, and such that the measure of a set is its density. These algebras 𝒜 are “abstract” in the sense that the countable join in the underlying lattice is not ordinary set union. However they are “concrete” in the sense that the elements of the algebra are sets, the notion of an integrable function is available and the normed vector space of integrable functions can be shown to be isometrically isomorphic to an ordinary L1 space. If a function f is integrable, it is shown that its integral is given by

    1 N∑
limN N-    f∗(j)

where f is a suitably chosen function differing from f only on a set of density 0.

This construction differs from others (several are described by Kubilius in his book on probabilistic methods in number theory), because usually countable additivity is sacrificed, whereas here the meaning of countable join has been altered.

Mathematical Subject Classification
Primary: 28.65
Received: 2 October 1969
Published: 1 September 1970
Richard A. Scoville