The paper deals with the L. E. Dubins and L. J. Savage [Inequalities for stochastic processes. How to gamble if you must. (1976; Zbl 0359.60002)] gambling problem \({\mathcal E}=(X,\Gamma,u)\), where X is a set, u: \(X\to {\mathbb{R}}\), while \(\Gamma\) maps X into sets of finitely additive measures on X; \({\mathcal E}\) is called analytic if X is a Borel set, the graph of \(\Gamma\) is an analytic set (in \(X\times Prob(X))\) while u is bounded and upper analytic (i.e. \(\{x| u(x)>\alpha \}\) is analytic for every real \(\alpha)\); u(\(\sigma\),t) denotes the expected utility under the choice of strategy \(\sigma =(\sigma_ 0,\sigma_ 1,...)\) (which means \(\sigma_ n(x_ 1,...,x_ n)\in \Gamma (x_ n)\) and stopping time t. The authors deal with the functions \[ V(x):=\sup \{\limsup_ tu(\sigma,t)| \quad \sigma \text{ is a strategy available at }x\} \] and \(V_ M(x)\) defined as above but with the choice of strategies restricted to universally measurable functions.
The main theorem of the paper, saying that \(V=V_ M\) and being upper analytic whenever \({\mathcal E}\) is analytic, generalizes a result of R. E. Strauch [Trans. Am. Math. Soc. 126, 64-72 (1967; Zbl 0166.157)]. The proof is based on a theorem of Y. N. Moschovakis [Descriptive set theory. (1980; Zbl 0433.03025), Theorem 7.C.8]. The authors also give alternative characterizations of V.