Distribution function and decreasing rearrangement 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?

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 $F(\alpha)<2^k$.
