Presumely you mean the "noncommutative integral" which is just the Dixmier trace. You can find its construction in Várilly's book, for example. It comes from the theory of pseudodifferential operators but to obtain more examples one can take Dixmier trace against an operator to make sure that the outcome is finite. So if this is your measure then is not supposed to be integrated but sometimes when evaluated on an operator it can be shown to coincide with an integral of the operator's "symbol".

Andreas is right, but let me put some background. A Riesz representation theorem says that the non negative linear functionals on the C^* algebra of continuous functions with compact support on a locally compact Hausdorff space can be identified with the space of Borel measures. Now a locally compact Hausdorf space is "essentially the same" as this algebra by the Gelfand-transform: the space of continuous non trivial homomorphisms to $\mathBB C$ (equivalently maximal ideals by the Gelfand Naimark theorem), with the topology induced as closed subset of the linear functionals of norm 1 is homeomorphic to the locally compact space we started with and vice versa every C^* algebra defines a locally compact space in this way. In fact the correspondence is even functorial: continuous homomorphisms correspond to continuous functions (this is another consequence of the Gelfand Naimark theorem)

This is the basis for non commutative geometry which simply defines a non commutative space as a C^* algebra. Naively an integral should correspond to a non negative linear functional, but here we want something "geometric". This means that we want invariance under conjugation at least by unitary elements, i.e. u in the C^* algebra such that uu* = u*u = 1. This means in particular that different choices of orthogonal bases for concrete C^* algebras such as the matrix algebra's and the bounded operators on (countable dimension) Hilberspace do not change the value. This is crucial for applications in physics like quantum statistical mechanics, where such C^* algebra approaches are used as an alternative to the . The "right" generalisation of a measure then turns out to be a :trace" (or if we allow infinite values a weight) on a C^* algebra, i.e. a function Tr: PosCone(C) \to \mathBB R such that:

1 Tr(x + y) = Tr(x) + Tr(y) x, y positive (i.e. of the form zz^*)

2. Tr(\lambda x) = \lambda Tr(x) for \lambda > 0 , x positive

3. Tr(x^*) = Tr(x)^* x positive

4 Tr(x) \ge 0 x positive

5 Tr(xy) = Tr(yx) x, y positive

of course 5 is automatic for commutative C^* algebra's.

All this is described in Connes thick book: Non Commutative geometry. He also explains that just as the most general functions that can be integrate are the measurable functions, which form a commutative von Neumann algebra, Traces extend to von Neumann algebras, which is why he calls the theory of von Neumann algebras, non commutative measure theory