This is a contribution to the recent and growing literature on the modeling of (un)awareness. The author addresses the issue of how to distinguish between events an individual is aware of, but assigns zero probability to, and events the individual is not aware of. The analysis is carried out in a framework that integrates the syntax-free structures introduced by \textit{A. Heifetz} et al. [J. Econ. Theory 130, No. 1, 78--94 (2006; Zbl 1141.91598)] and the Anscombe-Aumann approach to subjective expected utility. The author proposes a notion of ``awareness-dependent subjective expected utility'', provides a characterization of it, and shows that unawareness has behavioral implications that are different from those of probability zero belief.