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In a series of papers Tsirelson constructed from measure types of random setsand generalised random processes a new range of examples for continuous tensorproduct systems of Hilbert spaces introduced by Arveson for classifying$E_0$-semigroups. This paper establishes the converse: Each continuous tensorproduct systems of Hilbert spaces comes with measure types of distributions ofrandom (closed) sets in [0,1] or $R_+$. These measure types are stationary andfactorise over disjoint intervals. In a special case of this construction, thecorresponding measure type is an invariant of the product system and the rangeof the invariant is characterized.  Moreover, based on a detailed study of this kind of measure types, weconstruct for each stationary factorizing measure type a continuous tensorproduct systems of Hilbert spaces such that this measure type arises as thebefore mentioned invariant.  The measure types of the above described kind are connected withrepresentations of the corresponding $L^\infty$-spaces. This leads to directintegral representations of the elements of a given product system whichcombine well under tensor products. Using this structure in a constructive way,we can relate to any (type III) product system a product system of type $II_0$preserving isomorphy classes. Thus, the classification of type III productsystems reduces to that of type II ones.  Further, we show that all consistent measurable structures on an algebraiccontinuous tensor product systems of Hilbert spaces yield isomorphic productsystems. Thus the measurable structure of a continuous tensor product systemsof Hilbert spaces is essentially determined by its algebraic one.

Random sets and invariants for (type II) continuous tensor product systems of Hilbert spaces