Noncommutative measure and integration refers to the theory of weights, states, and traces on von Neumann algebras (Takesaki 1979 v. 2 p. 141).
References I. E. Segal. A noncommutative extension of abstract integration. Ann. of Math. (2), 57:401–457, 1953. MR # 14:991f, JSTOR collection. 2.0(2) Kostecki, R.P. (2013), W*-algebras and noncommutative integration, arXiv:1307.4818, Bibcode:2013arXiv1307.4818P. Takesaki, M. (1979), Theory of Operator Algebras I, II, III, ISBN 3-540-42248-X, ISBN 3-540-42914-X. ISBN 3-540-42913-1