All sets described are finite subsets of $\mathbb{R}$.
Given a measurable and integrable non-negative function $f$ over some measurable domain $A$, I'm trying to show $f$ is integrable iff the collection $\lbrace \mu(A_k)\rbrace$ is summable where $A_k=\lbrace x\in E\,|n\le f(x) \rbrace.$
I proved the reverse, but I'm stuck on $(\Rightarrow)$.
Since $E$ is measurable all of the $A_k's$ are bounded and measurable. $f$ is integrable-i.e., its integral is finite, so since $\int_A f=\sup\lbrace\int_A h\,| \rbrace$ each element of the set $\lbrace \int_A h\rbrace$ where $0\le h\le f$ and $h$ is simple is finite. How do I use this to describe the summability of the set $\lbrace \mu(A_k) \rbrace $?