2007 seminar talk: Lebesgue measure and the coin-tossing game

Talk held by Vladimir Kanovei (Institute for Information Transmission Problems, Moscow) at the KGRC seminar on 2007-11-29.


Given a set A of infinite dyadic sequences, we consider a game between G, the gambler, and C, the casino. C successively plays bits b_0,b_1,b_2,... , and C definitely loses if the infinite sequence b=<b_0,b_1,b_2,...> does NOT belong to A.
And G bets on every next move of C.
Beginning with the initial balance say $1, G can bet any amount less than the current balance on one of two possible moves of C (0 or 1), and if C makes that move then the balance accordingly increases by the amount of bet.
Otherwise the balance decreases.
The final outcome of the game can be defined in terms of the limit of the supremum of the balance values.
And it turns out that the existence of certain strategies for G and C characterizes the Lebesgue measure characteristics of the set A. In brief, the smaller A is the bigger gains Casino can guarantee.

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