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Spaces of Valuations
Author(s) -
HECKMANN REINHOLD
Publication year - 1996
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1996.tb49168.x
Subject(s) - topological space , mathematics , valuation (finance) , real valued function , cone (formal languages) , pure mathematics , dual space , locally convex topological vector space , regular polygon , space (punctuation) , function space , topological vector space , discrete mathematics , topology (electrical circuits) , combinatorics , computer science , geometry , algorithm , finance , economics , operating system
Valuations are measurelike functions mapping the open sets of a topological space X into positive real numbers. They can be classified into finite, point continuous, and Scott continuous valuations. We define corresponding spaces of valuations V f X ⊂ V p X ⊂ VX . The main results of the paper are that V p X is the soberification of V f X , and that V p X is the free sober locally convex topological cone over X . From this universal property, the notion of the integral of a real‐valued function over a Scott continuous valuation can be easily derived. The integral is used to characterize the spaces V p X and VX as dual spaces of certain spaces of real‐valued functions on X .