If $d_f (t) : (0, \infty) \rightarrow [0,\infty]$ prove $d_f (t) \leq \bigg( \frac{||{f}||_{L^p(X)}}{t} \bigg)^p.$

Question

Let $1 \leq p < \infty$ and suppose that $f \in L^{p}(X).$ Define the function $d_f (t) : (0, \infty) \rightarrow [0,\infty]$ by $$d_f(t)= \mu (\{ x \in X : |f(x)|>t\}).$$ For each $t>0,$ prove that $$d_f (t) \leq \bigg( \frac{||{f}||_{L^p(X)}}{t} \bigg)^p.$$

I'm stomped by this problem, I don't even know where to begin. Could someone help me begin to understand this problem?

Answer 1

Write $$\|f\|_p^p =\underbrace{\int_{[|f|\leq t]}|f|^p}_{\geq 0}+\int_{[|f|>t]}|f|^p \geq t^p d_f(t).$$Rearrange.