The choice of admissible trading strategies in mathematical modeling of financial markets is a delicate issue, going back to Harrison and Kreps [J. Econom. Theory, 20 (1979), pp. 381-408]. In the context of optimal portfolio selection with expected utility preferences this question has been the focus of considerable attention over the last 20 years. We propose a novel notion of admissibility that has many pleasant features-admissibility is characterized purely under the objective measure P; each admissible strategy can be approximated by simple strategies using a finite number of trading dates; the wealth of any admissible strategy is a supermartingale under all pricing measures; local boundedness of the price process is not required; neither strict monotonicity, strict concavity, nor differentiability of the utility function is necessary; the definition encompasses both the classical mean-variance preferences and the monotone expected utility. For utility functions finite on R, our class represents a minimal set containing simple strategies which also contains the optimizer, under conditions that are milder than the celebrated reasonable asymptotic elasticity condition on the utility function.