On Measurable Gambling Problems

The function Mg is well-defined when g is a bounded, Borel measurable function and it is shown below that, in this case, Mg is universally measurable. The function F9 is well-defined if g is bounded and finitary. If g is bounded, finitary, and Borel, then both functions are well-defined and seen to be equal. Thus a gambler can do just as well when restricted to measurable strategies for these problems. These results seem to contain most of the known results on the measurability of the return function and the adequacy of measurable strategies but the problem which motivated this research nevertheless remains open. That is, do good measurable strategies exist for measurable problems with a measurable utility function of the type studied by Dubins and Savage? (If so, the return function is universally measurable.) Some progress is made on this question. Let u be a bounded function on F and a a strategy. Then u(a) is defined to be lim supt.,, f u(ft) du, where the lim sup is over all stop rules t. It is shown below that for u and a measurable, it is equivalent