Let B([0,1)] be the Borel sets of [0,1] and let X be an infinite-dimensional Banach space. Let μ:B([0,1)]⇒X be a countably-additive vector measure, and let f:X⇒R be a function with f(0)=0. Let pNA be the Banach space spanned by polynomials of nonatomic real-valued measures. Sufficient conditions are obtained for the game fℝoslash;μ to be in pNA, and a formula is developed for the value of such games. Moreover, examples are given to illustrate why the seemingly obvious finite-dimensional approximations are not applicable.
