Strong measure zero and infinite games

We show that strong measure zero sets (in a σ -totally bounded metric space) can be characterized by the nonexistence of a winning strategy in a certain infinite game. We use this characterization to give a proof of the well known fact, originally conjectured by K. Prikry, that every dense G δ subset of the real line contains a translate of every strong measure zero set. We also derive a related result which answers a question of J. Fickett.