Lebesgue measure and gambling

Lebesgue measure of point sets is characterized in terms of the existence of various strategies in a certain coin-flipping game. ‘Rational’ and ‘discrete’ modifications of this game are investigated. We prove that if one of the players has a winning strategy in a game of this type depending on a given set $P\subseteq[0,1]$, then this set is measurable.
Bibliography: 11 titles.