The value of certain pNA games through infinite-dimensional Banach spaces

LetB(0, 1]) be the Borel sets of 0, 1] and letX be an infinite-dimensional Banach space. Let :B(0, 1])X be a countably-additive vector measure, and letf:XR be a function withf (0)=0. LetpNA be the Banach space spanned by polynomials of nonatomic realvalued measures. Sufficient conditions are obtained for the gamef ^o to be inpNA, 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.