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We generalize the notion of regularity of measures to quantum logics and then prove that each regular finitely additive state on a quantum logic is countably additive. Examples are given from measure theory and quantum mechanics. In this note, we establish a generalization of the well-known theorem of Alexandroff: Each bounded, regular and finitely additive measure on a compact Hausdorff space is countably additive [2, p. 138]. Let (E, 0 and each q EE C there exists p E VP with p 0 and q EE C there exists p' EE C for some p E V such that p' > q and o(p' A q') < e. (There exists p EE P such that p < q' and o(p' A q') < c.) Let VP' = {p': p E VP }. An element p EE P is called finitely-coverable if {P'1'P'2' * * } C 'P', V??P'k exists in C and p < V'-IP'k implies for some n that Vnk1p'k exists in e and p < Vn-1p'k. VP is called finitelycoverable if each element of VP is finitely-coverable. We now state the Received by the editors January 20, 1977 and, in revised form, April 25, 1977. AMS (MOS) subject classifications (1970). Primary 28A60, 8 1A12, 06A25, 46G99.

States on quantum logics and their connection with a theorem of Alexandroff