Let P(M) be the projection lattice of an arbitrary JBW-algebra or von Neumann algebra M. It is shown that the tracial states of M correspond by extension precisely to the subadditive probability measures on P(M). The analogous result for normal semifinite traces is also proved. A positive measure on the projection lattice P(M) of a JBW-algebra M is a function p: P(M) -> [0, oo] such that p(e + f) = p(e) + p(f) whenever ef = 0. Such a measure is said to be a probability measure if p(l) = 1. A

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... itive measure is said to be normal if p(ea) -> p(e) whenever ea / e and semifinite if for each projection e there is a net of projections ea / e with p(ea) < oo, for all a. A subadditive measure on P(M) is a positive measure such that p(e\/f)<p(e) + p(f), for alle, feP(M).