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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Noncommutative measure and integration refers to the theory of weights, states, and traces on von Neumann algebras (Takesaki 1979 v. 2 p. 141). If a von Neumann algebra acts on a Hilbert space containing a norm 1 vector v, then the functional a (av,v) is a normal state. This construction can be reversed to give an action on a Hilbert space from a normal state: this is the GNS construction for normal states. Any factor has a trace such that the trace of a non-zero projection is non-zero and the trace of a projection is infinite if and only if the projection is infinite. Such a trace is unique up to rescaling. For factors that are separable or finite, two projections are equivalent if and only if they have the same trace.