Games and Measures

Abstract

Suppose that 0 < α < 1, 0 < β < 1. Consider the following game by players Black and White. First, Black picks a compact real interval B₁ of length ℓ(B₁). Next, White picks a compact interval W₁ ⊂ B₁ of length ℓ(W₁) = αℓ(B₁). Then Black picks a compact interval B₂ ⊂ W₁ of length ℓ(B₂) = βℓ(W₁), etc. In this way, a nested sequence of compact intervals

$$ B_1 \supset W_1 \supset B_2 \supset W_2 \supset \ldots $$

is generated, with lengths

$$ \ell (B_k ) = (\alpha \beta )^{k - 1} \ell (B_1 ) and \ell (W_k ) = (\alpha \beta )^{k - 1} \alpha \ell (B_1 ) (k = 1,2,3, \ldots ). $$

It is clear that $\mathop \cap \limits_{k = 1}^\infty B_k = \mathop \cap \limits_{k = 1}^\infty W_k$ consists of a single point.