Hyperfinite Von Neumann games

Let ${}^1\text{V(}\text{R}\text{)]}$ be an ω₁-saturated enlargement in the sense of Keisler (1977) and let $\text{F}$ be a hyperfinite finite set in ¹$\text{N}$. Following the suggestion of Wesley (1971) we define a class of hyperfinite games of the form: $\text{Γ}{}_{\mathrm{<mtext>F</mtext>}}\text{(}{}^1\text{υ)=〈Φ,}\text{A}\text{(}\text{F}\text{),}{}^1\text{υ〉}$, and show that measure-theoretic analogues of the kernel and bargaining set exist in this nonstandard setting such that their standard parts Loeb-measurable measurable on the Loeb space generated by the internal ¹finitely additive measure $\text{u}{}_{\mathrm{<mtext>F</mtext>}}\text{:}\text{A}\text{(}\text{F}\text{)→}{}^1\text{R}{}_{\mathrm{+}}$.