As you can tell from the title, this is an exercise from Axler's measure theory book, and I am struggling with the following problem:
Suppose $b_1,b_2,\dots$ is a sequence of real numbers. Define $f: \mathbb{R} \rightarrow [0,\infty]$ by
$$f(x) = \left\{\begin{array}{cc}
\sum\limits_{k=1}^\infty \frac{1}{4^k|x-b_k|} & \text{ if } x \notin \{b_1,b_2,\dots\},\\
\infty & \text{ if } x \in \{b_1,b_2,\dots\}
\end{array}\right.$$
Prove that $|\{x \in \mathbb{R} : f(x) < 1\}| = \infty$.
I am not sure what to do. Can someone help, please?
Best Answer
Consider the sets $$ E_k = \{x : |x-b_k| > 2^{-k}\},\qquad k=1,2,3\dots,\\ E = \bigcap_{k=1}^\infty E_k . $$ If $x \in E_k$ then $$ |x-b_k| > 2^{-k} ,\\ 4^k|x-b_k| > 2^{k} ,\\ \frac{1}{4^k|x-b_k|} < 2^{-k} . $$ If $x \in E$ then $$ f(x) = \sum_{k=1}^\infty \frac{1}{4^k|x-b_k|} < \sum_{k=1}^\infty 2^{-k} = 1 . $$ What is the measure of $E$? To compute that, consider complements: $$ E_k^c = \{|x-b_k| \le 2^{-k}\},\qquad |E_k^c| = 2\cdot 2^{-k} ,\\ |E^c| = \left|\bigcup_{k=1}^\infty E_k^c\right| \le \sum_{k=1}^\infty |E_k^c| = \sum_{k=1}^\infty 2\cdot 2^{-k} < \infty , $$ and therefore $|E| = \infty$.