Nowadays measures on logics of projectors are studied intensively. In particular, the extension problem of a finite measure to the integral in von Neumann algebras is completely solved [E. Christensen, Commun. Math. Phys. 86, 529-538 (1982; Zbl 0507.46052); the author, Teor. Mat. Fiz. 45, No.2, 244-250 (1980; Zbl 0467.46047)]. In the present paper, this result is generalized to \(AW^*\)-algebras.