Distribution function and decreasing rearrangement

Question

Let $(X,dx)$ a measure space and $f\in L^p(X,\mathbb{C})$; let's define its distribution function $$F(\alpha)=meas(\{x\in X||f(x)|>\alpha\})$$ and the decreasing rearrangement $$\alpha_k=\inf\{\alpha>0|F(\alpha)<2^k\}$$ I have to prove the following $$\sum_{k:\alpha_k>\alpha}2^k\leq 2 F(\alpha)$$ Any suggestions?

Answer 1

For any nonnegative number $t$ we have $$\sum_{k : 2^k\le t } 2^k\leq 2 t$$ This is because the sum of infinite geometric progression on the left is twice its largest term, by the geometric sum formula.

To solve your problem, use the above as $$\sum_{k : F(\alpha)\ge 2^k } 2^k\leq 2 F(\alpha)$$ and observe that the inequality $\alpha_k>\alpha$ implies $\alpha$ does not satisfy $F(\alpha)<2^k$.