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This paper first introduces the so-called quasi-continuous random dynamical system (RDS) on a separable Banach space. The quasi-continuity is weaker than all the usual continuities and thus is easier to check in practice. We then establish a necessary and sufficient condition for the existence of random attractors for the quasi-continuous RDS. We also give a general method to obtain the random attractors for the RDS on the Banach space L q (D) for q 2. As an application, it is shown that the RDS generated by the stochastic reaction-diffusion equation possesses a finite-dimensional random attractor in L q (D) for any q 2, a comparison result of fractal dimensions under the different L q -norms is also obtained.

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