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Opportunistic spectrum access (OSA) that allows secondary users to independently search for and exploit instantaneous spectrum availability is considered. The design objective is to maximize the throughput of a secondary user while limiting the probability of colliding with primary users. Integrated in the joint design are three basic components: a spectrum sensor that identifies spectrum opportunities, a sensing strategy that determines which channels in the spectrum to sense, and an access strategy that decides whether to access based on potentially erroneous sensing outcomes. This joint design is formulated as a constrained partially observable Markov decision process (POMDP), and a separation principle is established. The separation principle reveals the optimality of myopic policies for the design of the spectrum sensor and the access strategy, leading to closed-form optimal solutions. Furthermore, it decouples the design of the sensing strategy from that of the spectrum sensor and the access strategy, and reduces the constrained POMDP to an unconstrained one. Numerical examples are provided to study the tradeoff between sensing time and transmission time, the interaction between the physical layer spectrum sensor and the MAC layer sensing and access strategies, and the robustness of the ensuing design to model mismatch.