  We study a game first introduced by Martin (actually we use a slight
variation of this game) which plays a role for measure analogous to the
Banach-Mazur game for category. We first present proofs for the basic
connections between this game and measure, and then use the game to prove
fundamental measure theoretic results such as Fubini's theorem, the
Borel-Cantelli lemma, and a general unfolding result for the game which gives,
for example, the measurability of $\boldsymbol{\Sigma}^1_1$ sets. We also use
the game to give a new, more constructive, proof of a strong form of the
R\'{e}nyi-Lamperti lemma, an important result in probability theory with many
applications to number theory. The proofs we give are all direct combinatorial
arguments using the game, and do not depend on known measure theoretic
arguments.
