A counterexample to the epsilon-delta criterion for Absolute Continuity of Measures

Question

Let $p>0$, and let $\mu$ be a Borel measure on $[0,\infty)$ defined by $\mu(E)=\int_Ex^pd\lambda$ where $\lambda$ denotes Lebesgue measure. Show that $\mu$ is absolutely continuous with respect to $\lambda$, but $\mu$ does not meet the epsilon-delta criterion for absolute continuity, namely for every $\epsilon>0$ there's a $\delta>0$ such that $\mu(E)<\epsilon$ whenever $\lambda(E)<\delta$.

I've managed to prove that $\mu$ is absolutely continuous with respect to $\lambda$, but I'm not sure how to approach the second part. I basically need to find a sequence of sets $E_n$ (in $B([1,\infty)$) such that $\lambda(E_n)\rightarrow 0$ but $\int_{E_n}x^pd\lambda$ does not go to zero. But I can't even think of such a sequence in the case where $p=1$, let alone the general case.

Answer 1

Hopefully the case $p=1$ will help:

$$\lambda([a,b])= b-a$$ and $$ \mu([a,b]) = \frac{1}{2}(b^2-a^2) = \frac{1}{2}(b-a)(b+a) $$

Taking, for example, $a=3^n$ and $b=3^n+\frac{1}{2^n}$ will give you sets $E_n$ which you can show the desired properties of.

Answer 2

The issue is more or less that $x^{p+1}$ is not a uniformly continuous function: it grows faster and faster as $x \to \infty$, which means that $\mu(I)$ can be large even when $\lambda(I)$ is small if $I$ is located far to the right.

Being more precise, consider $\delta>0$ arbitrary and then try computing $\mu([\delta^{-1/p},\delta^{-1/p}+\delta])$.