Finite measure of the set $\{x\in \mathbb{R}^n: |f(x)|>\lambda\}$

Question

Could someone tell me if the following property is true?

Let us consider $C\geq 0$, $p>1$ and $f:\mathbb{R^n}\to\mathbb{R}$ a function such that $\int_{K}|f(x)|\leq C \mu \left(K\right)^{\frac{p}{p-1}} $ for every meausrable (in the sense of Lebesgue) set $K$ where $\mu$ denotes Lebesgue measure.

Can we ensure that $\mu\left(\{x\in \mathbb{R^n}:|f(x)|>r\}\right)<\infty$ for every $r>0$?.

I have tried to write $\mu\left(\{x\in \mathbb{R^n}:|f(x)|>r\}\right)$ as $\int_{K}|f(x)|$ for a certain set $K$ but I has not got it.

Any suggestions would be welcome because I am very lost. Thanks.

Answer 1

Hint: Suppose that the measure of $A = \{x \in \mathbb R^n : |f(x)| > r\}$ is infinite for some $r$. Then, it has measurable subsets $K$ with arbitrary large $\mu(K)$.