The economics profession has been for a long time divided into two camps — ADs and ACs.
On the AD camp, there is me, claiming that a bit of exciting bizarreness brought to game theory by the axiom of choice is not worth the headache of checking measurability of strategies and the embarrassment caused by undetermined gale-stewart games; The Axiom of Determinacy, on the other hand, offers a paradise in which every function is measurable and every zero-sum game with perfect monitoring is determined. When pressed, we — that is to say, I — are willing to accept the axiom of countable choice (without which you say goodbye to pretty much all of analysis) and the axiom of dependent choice (which essentially says that if at every day you have an action that let you stay alive one day longer, then you have a strategy that makes you immortal). So, the AD camp should perhaps more accurately be called the ZF+DC+AD camp: Zermelo Frenkel + Dependent Choice + Axiom of Determinacy.
The AC guys insist of sticking to the axiom of choice, mainly, I suppose, because of finitely additive probabilities.
Well, here is a nice compromise, called ZFC+PD, suggested to me by Alexander Kechris (In fact, I was partially aware of it already through the work of Maitra and Sudderth — more on that in a moment,) which I hereby commit to accept if the AC-s will collaborate. PD stands for the axiom of projective determinacy. Projective sets are essentially everything you encounter in real life. Not only Borel sets are projective, but also the images of Borel sets under Borel maps (AKA analytic sets) and their complements, and their Borel images, etc. This is a nice sigma-algebra of sets, which is closed under more operation then Borel sets, and also admits nice uniformization theorems. Also, every projective map is measurable.
Projective determinacy is undecidable in ZFC, but from what I understand it is by now a rather acceptable addition to ZFC among sets theorists and logicians. Hence ZFC+PD.
AC-s get to keep their finitely additive probabilities; AD-s get to keep all the nice regularity and closure properties we like in our sets, every gale-stewart game which can be even vaguely described is determined. And, in ZFC+PD the optimal reward function in a Borel gambling house is a measurable function of your initial capital !
Nice, huh ?