Direct sum $\Rightarrow$ Direct Integral, Tensor product $\Rightarrow$?

Is there a way to define a tensor product over a measure space(=index set) with a continuous measure for Hilbert spaces?

For the sum we have the notion of a direct integral, here.

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I doubt there is such a thing. Out of curiosity: why would one want to have such a construction? I can see some (although not much) need for countable tensor products. The utility and applicability of such a construction eludes me. I ask this also in order to clarify your intentions: what kind of properties would you like this gadget to have? Do you want just something you could call a tensor product or do you have an application in mind? – t.b. Jul 26 '11 at 10:54
Not really an application, but some ideas what it'd be. In the theory of stochastic processes, the $\sigma$ algebra often refine with continuous time. For certain processes, I'd think that this can be interpreted as such an infinite tensor product. I want a purely abstract definition here not involving the notion of stochastic processes. – plusepsilon.de Jul 26 '11 at 12:06