Let $f\geq 0$ be a measurable function which is finite almost everywhere.
For each $k\in\mathbb{Z}$, define, $E_k=\lbrace x|f(x)>2^k\rbrace, F_k=\lbrace x|2^k\leq f(x)<2^{k+1}\rbrace$.
Is it true that $\sum_{k=-\infty}^{\infty}2^km(E_k)<\infty$ if and only if $\sum_{k=-\infty}^{\infty}2^km(F_k)<\infty$?