Measure, Integration & Real Analysis · Problem 2A.12

Measures — Sheldon Axler

Mathematics — Measure Theory · outer-measure-fat-cantor-type-set

Problem

Suppose

r_1, r_2, \ldots

is a sequence that contains every rational number. Let

F = \mathbf{R} \setminus \bigcup_{k=1}^\infty \left(r_k - \frac{1}{2^k}, r_k + \frac{1}{2^k}\right).

((a)) Show that

F

is a closed subset of

\mathbf{R}

. ((b)) Prove that if

I

is an interval contained in

F

, then

I

contains at most one element. ((c)) Prove that

|F| = \infty