This is a practice problem from Carothers p. 321.
Let $f$ be nonnegative and measurable. Prove that $\int f < \infty$ if and only if $\sum_{-\infty}^\infty 2^km(\{f > 2^k\}) < \infty .$
One thing I noticed right away was that $\int 2^k \chi_A = 2^km(\{f > 2^k\})$ where $A=\{f > 2^k \}$