I need to find the approximate limit of $f=\sum_{k=1}^{\infty}\chi_{[1/k, 1/k+2^{-k}]}$ at zero.
Definition: $f:\mathbb{R}^n\rightarrow\bar{\mathbb{R}}$ has approximate limit $\lambda$ at $x$ if $$\lim_{r\rightarrow 0}\frac{m_n(B(x, r)\cap\{|f-\lambda|>\epsilon\})}{m_n(B(x, r))}=0.$$
Clearly only options are $0$ or $1$. But the measure of the set close to zero where $f$ is one tends to zero. So I think the approximate limit should be $0$. Then I should estimate $$m_n(B(0, r)\cap \{|f|>\epsilon\})=m_n(B(0, r)\cap \{f=1\}).$$ The set where $f=1$ is the union $$\{f=1\}=\bigcup_{k=1}^{\infty}[\frac{1}{k}, \frac{1}{k}+2^{-k}].$$ Therefore I should estimate the measure of $$\bigcup_{k=1}^{\infty}[0, r]\cap[\frac{1}{k}, \frac{1}{k}+2^{-k}].$$ But I don't know how to proceed. Any help is welcome.