We consider a game in which the action set of each player is uncountable, and show that, from weak assumptions on the common prior, for any mixed strategy there exists a pure strategy that is approximately equivalent to it . The assumption of this result can be further weakened if we consider a purification of an Nash equilibrium. Combined with the existence theorem for a Nash equilibrium, we derive an existence theorem for  pure strategy approximated Nash equilibrium under sufficiently weak assumptions. All  pure strategies we derive in this paper can take a finite number of possible actions.
We consider a game in which the action set of each player is uncountable, and show that, from weak assumptions on the common prior, any mixed strategy has an approximately equivalent pure strategy . The assumption of this result can be further weakened if we consider the purification of a Nash equilibrium. Combined with the existence theorem for a Nash equilibrium, we derive an existence theorem for a pure strategy approximated Nash equilibrium under sufficiently weak assumptions. All of the pure strategies we derive in this paper can take a finite number of possible actions.
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there exists a pure strategy that is approximately equivalent to it
has an approximately equivalent pure strategy
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the purification of a
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