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: Let nu be a alpha-finite Borel measure on a separable metric space T, and let (X(t) t epsilon T) be a measurable alpha-stable process, O<a<2. Sample path integrals of a certain type arise in many situations, e.g. in multiple stochastic stable integration (Rosinski and Woyczyski 1987), in inversion formulae for the Fourier transform of stable noise (Cambanis 1988), in integral transformations between stationary and stationary increments stable processes (Cambanis and Maejima 1990) and others. It is important, therefore, to know exactly when the above integral is finite. Although much is known about this question, certain things appear to have been unknown in the case p<1 and even the known results are scattered in the literature and have never been put together, mainly because different cases have been handled using very different tools, varying from path order analysis to geometry of certain Banach spaces. As a result, researchers working with stable processes have had to justify in each case existence of sample path integrals (see Cambanis and Maejima (1990) for a recent example). It is our purpose in this paper, therefore, to give necessary and sufficient conditions for sample path integrability of stable processes in the case which has been to use form.

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