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In its present form, your question is not well-posed (and this helps to explain the apparent discrepancy between the posted responses). In the case of infinite dimensional spaces, there are many possible definitions of the tensor product, even for Banach spaces, and the answer to your question will depend on the choice of a particular one. As pointed out in the first answer, there will be no such representation if you regard the space of measures as a Banach space in the usual way. However, there are suitable concepts of tensor products for which a representation of the type you desire is valid. In the simpler case of compact metric cases, e.g., products of compact intervals, the appropriate category is that of Waelbroeck spaces, in the non compact case you will require a more baroque one, that of CoSaks spaces. Unfortunately, the rather obscure nature of these constructions means that there is no explicit version to be found in the literature, to my knowledge, but it is quite easy to create one from known results in the Banach space situation.