Construct a measurable subset E such that $\lim_{\delta \to 0}\frac{1}{2\delta}\int_{p+\delta}^{p-\delta}\chi_E(x)dx=a$

Question

Let $0<a<1$ be a fixed constant . Construct a measurable subset E of $\mathbb{R}$ such that , for some $p\in E$ , we have $$\lim_{\delta \to 0}\frac{1}{2\delta}\int_{p+\delta}^{p-\delta}\chi_E(x)dx=a$$

Answer 1

you can build it by parts, Just make sure that the measure of the set in the intervals $(p+\frac{1}{n+1},p+\frac{1}{n})$ and $(p-\frac{1}{n+1},p-\frac{1}{n})$ is equal to $a(\frac{1}{n}-\frac{1}{n+1})=\frac{a}{n(n+1})$.

Since $\lim\frac{1}{n(n+1)}/{\frac{1}{n+1}}=0$ this set will satisfy $\lim_{\delta \to 0}\frac{1}{2\delta}\int_{p+\delta}^{p-\delta}\chi_E(x)dx=a$