Before revision:  The existence of such a scheme, whose practical value is tempered by its use of the Axiom of Choice , shows that imposing regularity conditions (such as measurability) is essential even in the foundations of game-theoretic probability .
After revision:  For simplicity the interest rate is assumed zero. If the price path is assumed continuous, the trader can become infinitely rich immediately after it becomes non-constant (if it ever does). If it is assumed positive, he can become infinitely rich immediately after reaching a point in time such that the variation of the log price is infinite in any right neighbourhood of that point (whereas reaching a point in time such that the variation of the log price is infinite in any left neighbourhood of that point is not sufficient). The practical value of these schemes is tempered by its use of the Axiom of Choice , shows that imposing regularity conditions (such as measurability) is essential even in the foundations of game-theoretic probability .