Measure, Integration & Real Analysis · Problem 2E.14

Integration — Sheldon Axler

Mathematics — Measure Theory · borel-function-infinite-set

Problem

Suppose b1,b2,…b_1, b_2, \ldots is a sequence of real numbers. Define f:R→[0,∞]f: \mathbf{R} \to [0, \infty] by f(x)={∑k=1∞14k∣x−bk∣if x∉{b1,b2,…},∞if x∈{b1,b2,…}.f(x) = \begin{cases} \displaystyle\sum_{k=1}^{\infty} \frac{1}{4^k |x - b_k|} & \text{if } x \notin \{b_1, b_2, \ldots\}, \\ \infty & \text{if } x \in \{b_1, b_2, \ldots\}. \end{cases} Prove that ∣{x∈R:f(x)<1}∣=∞|\{x \in \mathbf{R} : f(x) < 1\}| = \infty. [This exercise is a variation of a problem originally considered by Borel. If b1,b2,…b_1, b_2, \ldots contains all the rational numbers, then it is not even obvious that {x∈R:f(x)<∞}≠∅\{x \in \mathbf{R} : f(x) < \infty\} \neq \emptyset.]

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