I am trying to solve the following problem:
Let $\lambda_f(t) := \mu(\{x \in X: |f(x)| > t\})$
Prove that $f \in L^p(\mathbb R^d)$ for $0<p<\infty$ if and only if $$\sum_{k=-\infty}^{k=\infty}2^{kp}\lambda_f(2^k)<\infty$$ Show that there exist constant $c_1,c_2>0$ with $$c_1(\sum_{k=-\infty}^{k=\infty}2^{kp}\lambda_f(2^k))^{\frac{1}{p}}\leq ||f||_p \leq c_2(\sum_{k=-\infty}^{k=\infty}2^{kp}\lambda_f(2^k))^{\frac{1}{p}}$$
I am a bit lost with the series going from minus infinity, I'll write what I could do:
Notice that $\mathbb R^d$ can be expressed as the disjoint union of the following sets:$$E_k=\{x: 2^k<|f(x)|\leq 2^{k+1}\}$$
And that if $S_k=\bigcup_{j=k}^{\infty}E_j$, then $$\sum_{k=-\infty}^{k=\infty}2^{kp}\chi_{S_k}(x) \leq |f(x)|^p$$
If I could interchange this series with the integral, then I would have $$\sum_{k=-\infty}^{k=\infty}2^{kp}\lambda_f(2^k)<\int_{\mathbb R^d} |f(x)|^pdx,$$
from this inequality one can deduce $f \in L^p$ implies that the series is convergent.
I don't know how to justify that I can interchange the integral with the series and I don't know how to prove the last two inequalities. Actually, if I could show directly the last two inequalities, the if and only if follows directly from there.
Any help would be greatly appreciated. Thanks in advance.