4
$\begingroup$

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 $A=\{f > 2^k \}$

  • 0
    You're right. I just noticed that I left off the $<\infty$2011-09-24

1 Answers 1