Vol. 39, No. 2, 1971

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Measures on semilattices

Solomon Leader

Vol. 39 (1971), No. 2, 407–423

New definitions are given for positivity and bounded variation of functions on a semilattice S so that such functions extend to measures (respectively, signed measures) on the σ-algebra generated by some representation of S as a semilattice of sets under intersection. All such representations lie in a Stone space determined by S. Functions on a subsemilattice S of a distributive lattice L which extend to isotone valuations on L are characterized in terms of a partial ordering of finite sequences in S. Functions on a regulated semilattice which correspond to regular Borel measures on the associated locally compact space are characterized in terms of inclusion-exclusion sums.

Mathematical Subject Classification 2000
Primary: 28A60
Received: 4 November 1970
Published: 1 November 1971
Solomon Leader