Absolute continuity: For all $\epsilon >0$ there exists a $\delta >0$ such that if $\mu (A)<\delta $, ${\int}_{A}|f|d\mu <\u03f5$. Here $A$ is a measurable subset of $E$.

I know that if $f$ is integrable then it is absolutely continuous. But is there anyway I can show that absolute continuity implies integrability when $\mu $ is finite?

$\int |f|d\mu ={\int}_{A}|f|d\mu +{\int}_{{A}^{c}}|f|d\mu $

Can I choose some special set $A$ such that $\mu (A)<\delta $?

I know that if $f$ is integrable then it is absolutely continuous. But is there anyway I can show that absolute continuity implies integrability when $\mu $ is finite?

$\int |f|d\mu ={\int}_{A}|f|d\mu +{\int}_{{A}^{c}}|f|d\mu $

Can I choose some special set $A$ such that $\mu (A)<\delta $?